<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-version="1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="False" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps4"> <h2 class="hd hd-2 unit-title">Problem set 4</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@ps4_hashpdf"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-version="1" data-block-type="html" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="False" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@ps4_hashpdf"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p>You can download this pdf file from <a href="/assets/courseware/v1/13ce40531b7054780fa259207a5d10b5/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps4.pdf">here</a></p><iframe src="/assets/courseware/v1/13ce40531b7054780fa259207a5d10b5/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps4.pdf" width="100%" height="1000px"> </iframe> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-version="1" data-block-type="vertical" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="VerticalStudentView" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="False" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@ps4-checkvertical"> <h2 class="hd hd-2 unit-title">Pset 4 problem checker</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-1-check" class="problems-wrapper" role="group" aria-labelledby="ps4-1-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-1-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 1 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-1-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; To check your answers put them in the appropriate box and click the &amp;#8216;Check&#39; button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. &lt;/p&gt; &lt;p&gt; Unless otherwise stated the checker expects numerical answers to 3 significant digits. &lt;/p&gt; &lt;p&gt; For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] &lt;b class=&#34;bf&#34;&gt;NOT &lt;/b&gt; [mathjaxinline]3x[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. &lt;/p&gt; &lt;p&gt; The formula checker expects a specific letter for the variables, so be sure to use the ones specified. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(a)&lt;/b&gt; Using the variables &lt;tt class=&#34;tt&#34;&gt;x&lt;/tt&gt; and &lt;tt class=&#34;tt&#34;&gt;lambda&lt;/tt&gt;, give the formula for [mathjaxinline]P(X \geq x)[/mathjaxinline]. &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_2_1&#34; id=&#34;input_ps4-1-check_2_1&#34; aria-describedby=&#34;status_ps4-1-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_2_1_dynamath&#34; name=&#34;input_ps4-1-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(b)&lt;/b&gt; Use the variable &lt;tt class=&#34;tt&#34;&gt;lambda&lt;/tt&gt; to give each of the values asked for &lt;/p&gt; &lt;p&gt; [mathjaxinline]E[X][/mathjaxinline] = &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_3_1&#34; id=&#34;input_ps4-1-check_3_1&#34; aria-describedby=&#34;status_ps4-1-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_3_1_dynamath&#34; name=&#34;input_ps4-1-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\text {Var}(X)[/mathjaxinline] = &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_4_1&#34; id=&#34;input_ps4-1-check_4_1&#34; aria-describedby=&#34;status_ps4-1-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_4_1_dynamath&#34; name=&#34;input_ps4-1-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\sigma _ X[/mathjaxinline] = &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_5_1&#34; id=&#34;input_ps4-1-check_5_1&#34; aria-describedby=&#34;status_ps4-1-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_5_1_dynamath&#34; name=&#34;input_ps4-1-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(c)&lt;/b&gt; Using the variables &lt;tt class=&#34;tt&#34;&gt;t&lt;/tt&gt; and &lt;tt class=&#34;tt&#34;&gt;lambda&lt;/tt&gt;, give the formulas for the cdf [mathjaxinline]F_ T(t)[/mathjaxinline] and pdf [mathjaxinline]f_ T(t)[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; [mathjaxinline]F_ T(t) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_6_1&#34; id=&#34;input_ps4-1-check_6_1&#34; aria-describedby=&#34;status_ps4-1-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_6_1_dynamath&#34; name=&#34;input_ps4-1-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]f_ T(t) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_7_1&#34; id=&#34;input_ps4-1-check_7_1&#34; aria-describedby=&#34;status_ps4-1-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_7_1_dynamath&#34; name=&#34;input_ps4-1-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(d)&lt;/b&gt; Give the expected time before one of the bulbs fails. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 7&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-1-check_8_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-1-check_8_1&#34; id=&#34;input_ps4-1-check_8_1&#34; aria-describedby=&#34;status_ps4-1-check_8_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-1-check_8_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-1-check_8_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-1-check_8_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-1-check_8_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-1-check_8_1_dynamath&#34; name=&#34;input_ps4-1-check_8_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 1&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-1-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-1-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-1-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-1-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-1-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-2-check" class="problems-wrapper" role="group" aria-labelledby="ps4-2-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-2-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-2-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; For this problem give your estimates to 2 decimal places. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(a)&lt;/b&gt; Give the probability at least 52.5% prefer Alessandre. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-2-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-2-check_2_1&#34; id=&#34;input_ps4-2-check_2_1&#34; aria-describedby=&#34;status_ps4-2-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-2-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-2-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-2-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-2-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-2-check_2_1_dynamath&#34; name=&#34;input_ps4-2-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;2(b)&lt;/b&gt; Give the probability that less than 31% prefer Gabriel, Sarah or So Hee &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-2-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-2-check_3_1&#34; id=&#34;input_ps4-2-check_3_1&#34; aria-describedby=&#34;status_ps4-2-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-2-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-2-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-2-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-2-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-2-check_3_1_dynamath&#34; name=&#34;input_ps4-2-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-2-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-2-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-2-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-2-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-2-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-3-check" class="problems-wrapper" role="group" aria-labelledby="ps4-3-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-3-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 3 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-3-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3.&lt;/b&gt; Give your estimate that the total rounding error exceeds 100 in absolute value. (Give the answer to 2 significant digits.) &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-3-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-3-check_2_1&#34; id=&#34;input_ps4-3-check_2_1&#34; aria-describedby=&#34;status_ps4-3-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-3-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-3-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-3-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-3-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-3-check_2_1_dynamath&#34; name=&#34;input_ps4-3-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 3&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-3-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-3-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-3-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-3-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-3-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-4a-check" class="problems-wrapper" role="group" aria-labelledby="ps4-4a-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-4a-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4a &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4a-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(a)&lt;/b&gt; Give the range of [mathjaxinline]X[/mathjaxinline] as an interval. You can use &lt;tt class=&#34;tt&#34;&gt;infty&lt;/tt&gt; if one of the bounds is infinity. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4a-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4a-check_2_1&#34; id=&#34;input_ps4-4a-check_2_1&#34; aria-describedby=&#34;status_ps4-4a-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4a-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4a-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4a-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4a-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4a-check_2_1_dynamath&#34; name=&#34;input_ps4-4a-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the pdf [mathjaxinline]f_ X(x)[/mathjaxinline] for [mathjaxinline]X[/mathjaxinline]. Use the variable x in your answer. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4a-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4a-check_3_1&#34; id=&#34;input_ps4-4a-check_3_1&#34; aria-describedby=&#34;status_ps4-4a-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4a-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4a-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4a-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4a-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4a-check_3_1_dynamath&#34; name=&#34;input_ps4-4a-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the range of [mathjaxinline]Y[/mathjaxinline]: &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4a-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4a-check_4_1&#34; id=&#34;input_ps4-4a-check_4_1&#34; aria-describedby=&#34;status_ps4-4a-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4a-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4a-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4a-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4a-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4a-check_4_1_dynamath&#34; name=&#34;input_ps4-4a-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the pdf [mathjaxinline]f_ Y(y)[/mathjaxinline] for [mathjaxinline]Y[/mathjaxinline]. Use the variable y in your answer. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4a-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4a-check_5_1&#34; id=&#34;input_ps4-4a-check_5_1&#34; aria-describedby=&#34;status_ps4-4a-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4a-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4a-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4a-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4a-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4a-check_5_1_dynamath&#34; name=&#34;input_ps4-4a-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4a&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-4a-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-4a-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4a-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4a-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4a-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-4b-check" class="problems-wrapper" role="group" aria-labelledby="ps4-4b-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-4b-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4b &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4b-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(b)&lt;/b&gt; Give the range of [mathjaxinline]W[/mathjaxinline]. You can use the variables &lt;tt class=&#34;tt&#34;&gt;a&lt;/tt&gt; and &lt;tt class=&#34;tt&#34;&gt;b&lt;/tt&gt; in your answer. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4b-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4b-check_2_1&#34; id=&#34;input_ps4-4b-check_2_1&#34; aria-describedby=&#34;status_ps4-4b-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4b-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4b-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4b-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4b-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4b-check_2_1_dynamath&#34; name=&#34;input_ps4-4b-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the pdf [mathjaxinline]f_ W(w)[/mathjaxinline] of [mathjaxinline]W[/mathjaxinline]. Use the variables &lt;tt class=&#34;tt&#34;&gt;w&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;a&lt;/tt&gt; and &lt;tt class=&#34;tt&#34;&gt;b&lt;/tt&gt; in your answer. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4b-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4b-check_3_1&#34; id=&#34;input_ps4-4b-check_3_1&#34; aria-describedby=&#34;status_ps4-4b-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4b-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4b-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4b-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4b-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4b-check_3_1_dynamath&#34; name=&#34;input_ps4-4b-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4b&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-4b-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-4b-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4b-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4b-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4b-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-5" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-4c-check" class="problems-wrapper" role="group" aria-labelledby="ps4-4c-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-4c-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4c &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-4c-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(c)&lt;/b&gt; Give the range of [mathjaxinline]V[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4c-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4c-check_2_1&#34; id=&#34;input_ps4-4c-check_2_1&#34; aria-describedby=&#34;status_ps4-4c-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4c-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4c-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4c-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4c-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4c-check_2_1_dynamath&#34; name=&#34;input_ps4-4c-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the pdf [mathjaxinline]f_ V(v)[/mathjaxinline] of [mathjaxinline]V[/mathjaxinline]. Use the variable &lt;tt class=&#34;tt&#34;&gt;v&lt;/tt&gt; in your answer. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-4c-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-4c-check_3_1&#34; id=&#34;input_ps4-4c-check_3_1&#34; aria-describedby=&#34;status_ps4-4c-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-4c-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-4c-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-4c-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-4c-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-4c-check_3_1_dynamath&#34; name=&#34;input_ps4-4c-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4c&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-4c-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-4c-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4c-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4c-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-4c-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-6" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-5a-check" class="problems-wrapper" role="group" aria-labelledby="ps4-5a-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-5a-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 5a &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5a-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;5(a)&lt;/b&gt; Note: The checker will not check units. It expects your numerical values to be in the correct units. On your paper, don&#39;t forget that you need to give the units explicitly. &lt;/p&gt; &lt;p&gt; Give each of the following. &lt;/p&gt; &lt;p&gt; [mathjaxinline]E[X] =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_2_1&#34; id=&#34;input_ps4-5a-check_2_1&#34; aria-describedby=&#34;status_ps4-5a-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_2_1_dynamath&#34; name=&#34;input_ps4-5a-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\text {Var}(X) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_3_1&#34; id=&#34;input_ps4-5a-check_3_1&#34; aria-describedby=&#34;status_ps4-5a-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_3_1_dynamath&#34; name=&#34;input_ps4-5a-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\sigma _ X =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_4_1&#34; id=&#34;input_ps4-5a-check_4_1&#34; aria-describedby=&#34;status_ps4-5a-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_4_1_dynamath&#34; name=&#34;input_ps4-5a-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]E[Y] =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_5_1&#34; id=&#34;input_ps4-5a-check_5_1&#34; aria-describedby=&#34;status_ps4-5a-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_5_1_dynamath&#34; name=&#34;input_ps4-5a-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\text {Var}(Y) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_6_1&#34; id=&#34;input_ps4-5a-check_6_1&#34; aria-describedby=&#34;status_ps4-5a-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_6_1_dynamath&#34; name=&#34;input_ps4-5a-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\sigma _ Y =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_7_1&#34; id=&#34;input_ps4-5a-check_7_1&#34; aria-describedby=&#34;status_ps4-5a-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_7_1_dynamath&#34; name=&#34;input_ps4-5a-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; For [mathjaxinline]W[/mathjaxinline], you can use the variables &lt;tt class=&#34;tt&#34;&gt;a&lt;/tt&gt;, &lt;tt class=&#34;tt&#34;&gt;b&lt;/tt&gt; in your answers. &lt;/p&gt; &lt;p&gt; [mathjaxinline]E[W] =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 7&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_8_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_8_1&#34; id=&#34;input_ps4-5a-check_8_1&#34; aria-describedby=&#34;status_ps4-5a-check_8_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_8_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_8_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_8_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_8_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_8_1_dynamath&#34; name=&#34;input_ps4-5a-check_8_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\text {Var}(W) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 8&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_9_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_9_1&#34; id=&#34;input_ps4-5a-check_9_1&#34; aria-describedby=&#34;status_ps4-5a-check_9_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_9_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_9_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_9_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_9_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_9_1_dynamath&#34; name=&#34;input_ps4-5a-check_9_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]\sigma _ W =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 9&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5a-check_10_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5a-check_10_1&#34; id=&#34;input_ps4-5a-check_10_1&#34; aria-describedby=&#34;status_ps4-5a-check_10_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5a-check_10_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5a-check_10_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5a-check_10_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5a-check_10_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5a-check_10_1_dynamath&#34; name=&#34;input_ps4-5a-check_10_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 5a&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-5a-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-5a-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5a-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5a-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5a-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-7" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-5b-check" class="problems-wrapper" role="group" aria-labelledby="ps4-5b-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-5b-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 5b &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5b-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;5(b)&lt;/b&gt; Give [mathjaxinline]E[V][/mathjaxinline]. In this case the checker expects an exact answer. &lt;/p&gt; &lt;p&gt; [mathjaxinline]E[V] =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5b-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5b-check_2_1&#34; id=&#34;input_ps4-5b-check_2_1&#34; aria-describedby=&#34;status_ps4-5b-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5b-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5b-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5b-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5b-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5b-check_2_1_dynamath&#34; name=&#34;input_ps4-5b-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 5b&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-5b-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-5b-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5b-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5b-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5b-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-8" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-5c-check" class="problems-wrapper" role="group" aria-labelledby="ps4-5c-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-5c-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 5c &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-5c-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;5(c)&lt;/b&gt; Give your answers to 3 significant digits. &lt;/p&gt; &lt;p&gt; Median of [mathjaxinline]X =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5c-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5c-check_2_1&#34; id=&#34;input_ps4-5c-check_2_1&#34; aria-describedby=&#34;status_ps4-5c-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5c-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5c-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5c-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5c-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5c-check_2_1_dynamath&#34; name=&#34;input_ps4-5c-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Median of [mathjaxinline]V =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-5c-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-5c-check_3_1&#34; id=&#34;input_ps4-5c-check_3_1&#34; aria-describedby=&#34;status_ps4-5c-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-5c-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-5c-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-5c-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-5c-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-5c-check_3_1_dynamath&#34; name=&#34;input_ps4-5c-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 5c&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-5c-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-5c-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5c-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5c-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-5c-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-9" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-version="1" data-block-type="problem" data-runtime-class="LmsRuntime" data-request-token="4d8cf486b23411efb51f16fffa68ec97" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps4-6-check" class="problems-wrapper" role="group" aria-labelledby="ps4-6-check-problem-title" data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps4-6-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 6 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps4-6-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Give your answers to 2 significant figures. The checker will understand scientific notation, e.g. 1.23e-4. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(a)&lt;/b&gt; Give the probability IQ is greater than 160. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_2_1&#34; id=&#34;input_ps4-6-check_2_1&#34; aria-describedby=&#34;status_ps4-6-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_2_1_dynamath&#34; name=&#34;input_ps4-6-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(b)&lt;/b&gt; Assuming a normal distribution, give the probability Modified_IQ is more than [mathjaxinline]4\sigma[/mathjaxinline] above the mean. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_3_1&#34; id=&#34;input_ps4-6-check_3_1&#34; aria-describedby=&#34;status_ps4-6-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_3_1_dynamath&#34; name=&#34;input_ps4-6-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(c)&lt;/b&gt; Assuming a t-distribution, give the probability Modified_IQ is more than [mathjaxinline]4\sigma[/mathjaxinline] above the mean. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_4_1&#34; id=&#34;input_ps4-6-check_4_1&#34; aria-describedby=&#34;status_ps4-6-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_4_1_dynamath&#34; name=&#34;input_ps4-6-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;6(d)&lt;/b&gt; We&#39;ll just tell you that all 3 normal probabilities are less than [mathjaxinline]10^{-30}[/mathjaxinline]. So we&#39;ll accept the answer 0 &amp;#8211;which is what R will tell you. &lt;/p&gt; &lt;p&gt; Give the t-distribution probabilities: &lt;/p&gt; &lt;p&gt; [mathjaxinline]P([/mathjaxinline]Modified_IQ [mathjaxinline]&amp;gt; 20) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_5_1&#34; id=&#34;input_ps4-6-check_5_1&#34; aria-describedby=&#34;status_ps4-6-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_5_1_dynamath&#34; name=&#34;input_ps4-6-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]P([/mathjaxinline]Modified_IQ [mathjaxinline]&amp;gt; 40) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_6_1&#34; id=&#34;input_ps4-6-check_6_1&#34; aria-describedby=&#34;status_ps4-6-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_6_1_dynamath&#34; name=&#34;input_ps4-6-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; [mathjaxinline]P([/mathjaxinline]Modified_IQ [mathjaxinline]&amp;gt; 200) =[/mathjaxinline] &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps4-6-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps4-6-check_7_1&#34; id=&#34;input_ps4-6-check_7_1&#34; aria-describedby=&#34;status_ps4-6-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps4-6-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps4-6-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps4-6-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps4-6-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps4-6-check_7_1_dynamath&#34; name=&#34;input_ps4-6-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 6&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps4-6-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps4-6-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-6-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-6-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps4-6-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>