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<h2 class="hd hd-2 unit-title">Problem set 5</h2>
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<h2 class="hd hd-2 unit-title">Pset 5 problem checker</h2>
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<h3 class="hd hd-3 problem-header" id="ps5-1ab-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-1ab-check-problem-progress" tabindex="-1">
Problem 1ab
</h3>
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<div class="problem">
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<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you can use R notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions, e.g. use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the ones specified. </p>
<p><b class="bf">1(a)</b> Give the value of [mathjaxinline]c[/mathjaxinline]. </p>
<p>
[mathjaxinline]c =[/mathjaxinline]<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-1ab-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_2_1" id="input_ps5-1ab-check_2_1" aria-describedby="status_ps5-1ab-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_2_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_2_1" class="answer"/>
<div id="display_ps5-1ab-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_2_1_dynamath" name="input_ps5-1ab-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
For the rest of this problem you can use <tt class="tt">c</tt> instead of the numerical value found in part (a). </p>
<p>
Give the formula for [mathjaxinline]F(x,y)[/mathjaxinline] Use <tt class="tt">x, y</tt> as the variables. </p>
<p>
[mathjaxinline]F(x,y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-1ab-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_3_1" id="input_ps5-1ab-check_3_1" aria-describedby="status_ps5-1ab-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_3_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_3_1" class="answer"/>
<div id="display_ps5-1ab-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_3_1_dynamath" name="input_ps5-1ab-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">1(b)</b> Give the marginal pdfs. Use the variable indicated </p>
<p>
[mathjaxinline]f_ X(x) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-1ab-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_4_1" id="input_ps5-1ab-check_4_1" aria-describedby="status_ps5-1ab-check_4_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_4_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_4_1" class="answer"/>
<div id="display_ps5-1ab-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_4_1_dynamath" name="input_ps5-1ab-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]f_ Y(y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps5-1ab-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_5_1" id="input_ps5-1ab-check_5_1" aria-describedby="status_ps5-1ab-check_5_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_5_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_5_1" class="answer"/>
<div id="display_ps5-1ab-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_5_1_dynamath" name="input_ps5-1ab-check_5_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the marginal cdfs. Use the variable indicated </p>
<p>
[mathjaxinline]F_ X(x) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps5-1ab-check_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_6_1" id="input_ps5-1ab-check_6_1" aria-describedby="status_ps5-1ab-check_6_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_6_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_6_1" class="answer"/>
<div id="display_ps5-1ab-check_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_6_1_dynamath" name="input_ps5-1ab-check_6_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]F_ Y(y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_ps5-1ab-check_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1ab-check_7_1" id="input_ps5-1ab-check_7_1" aria-describedby="status_ps5-1ab-check_7_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-1ab-check_7_1"/>
<span class="status unanswered" id="status_ps5-1ab-check_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1ab-check_7_1" class="answer"/>
<div id="display_ps5-1ab-check_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1ab-check_7_1_dynamath" name="input_ps5-1ab-check_7_1_dynamath"/>
</div>
</div></div> </p>
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aria-labelledby="ps5-1cde-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-1cde-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-1cde-check/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ps5-1cde-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-1cde-check-problem-progress" tabindex="-1">
Problem 1cde
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-1cde-check-problem-progress"></div>
<div class="problem">
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<p><b class="bf">1(c)</b> Give the values of [mathjaxinline]E[X][/mathjaxinline] and [mathjaxinline]\text {Var}(X)[/mathjaxinline]. (Use at least 3 significant digits. An exact fraction will also work.) </p>
<p>
[mathjaxinline]E[X] =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-1cde-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1cde-check_2_1" id="input_ps5-1cde-check_2_1" aria-describedby="status_ps5-1cde-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-1cde-check_2_1"/>
<span class="status unanswered" id="status_ps5-1cde-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1cde-check_2_1" class="answer"/>
<div id="display_ps5-1cde-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1cde-check_2_1_dynamath" name="input_ps5-1cde-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]\text {Var}(X) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-1cde-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1cde-check_3_1" id="input_ps5-1cde-check_3_1" aria-describedby="status_ps5-1cde-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-1cde-check_3_1"/>
<span class="status unanswered" id="status_ps5-1cde-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1cde-check_3_1" class="answer"/>
<div id="display_ps5-1cde-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1cde-check_3_1_dynamath" name="input_ps5-1cde-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">1(d)</b> Give the covariance and correlation of [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline]. (Use at least 2 significant digits.) </p>
<p>
[mathjaxinline]\text {Cov}(X,Y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-1cde-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1cde-check_4_1" id="input_ps5-1cde-check_4_1" aria-describedby="status_ps5-1cde-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-1cde-check_4_1"/>
<span class="status unanswered" id="status_ps5-1cde-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1cde-check_4_1" class="answer"/>
<div id="display_ps5-1cde-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1cde-check_4_1_dynamath" name="input_ps5-1cde-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]\text {Cor}(X,Y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps5-1cde-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-1cde-check_5_1" id="input_ps5-1cde-check_5_1" aria-describedby="status_ps5-1cde-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-1cde-check_5_1"/>
<span class="status unanswered" id="status_ps5-1cde-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-1cde-check_5_1" class="answer"/>
<div id="display_ps5-1cde-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-1cde-check_5_1_dynamath" name="input_ps5-1cde-check_5_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">1(e)</b> The checker will basically give you the answer here. Remember that on your paper you need to give a clear justification for the answer. </p>
<p>
Are [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] independent?<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps5-1cde-check_6_1">
<fieldset aria-describedby="status_ps5-1cde-check_6_1">
<div class="field">
<input type="radio" name="input_ps5-1cde-check_6_1" id="input_ps5-1cde-check_6_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps5-1cde-check_6_1-choice_1-label" for="input_ps5-1cde-check_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps5-1cde-check_6_1"> <text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps5-1cde-check_6_1" id="input_ps5-1cde-check_6_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps5-1cde-check_6_1-choice_2-label" for="input_ps5-1cde-check_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps5-1cde-check_6_1"> <text> No</text>
</label>
</div>
<span id="answer_ps5-1cde-check_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps5-1cde-check_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div> </p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1cde" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps5-1cde-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-1cde-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-1cde-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-1cde-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-1cde-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="35dcc5e0075811f0a1fb0affe2bbc7c1" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-2-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-2-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-2-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-2-check-problem-progress"></div>
<div class="problem">
<div>
<b class="bf">2.</b>
<em>No answer checker given.</em>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 2" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps5-2-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-2-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-2-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-2-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-2-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="35dcc5e0075811f0a1fb0affe2bbc7c1" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-has-score="True" data-graded="False" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ps5-3-check" class="problems-wrapper" role="group"
aria-labelledby="ps5-3-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ps5-3-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-3-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">2.</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">3(a)</b> Give each of the probabilities in the joint probability table for [mathjaxinline]p(x,y)[/mathjaxinline]. Your answers should be formulas using the parameter <tt class="tt">c</tt>. </p>
<p>
[mathjaxinline]p(1,1) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_2_1" id="input_ps5-3-check_2_1" aria-describedby="status_ps5-3-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_2_1"/>
<span class="status unanswered" id="status_ps5-3-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_2_1" class="answer"/>
<div id="display_ps5-3-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_2_1_dynamath" name="input_ps5-3-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]p(1,-1) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-3-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_3_1" id="input_ps5-3-check_3_1" aria-describedby="status_ps5-3-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_3_1"/>
<span class="status unanswered" id="status_ps5-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_3_1" class="answer"/>
<div id="display_ps5-3-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_3_1_dynamath" name="input_ps5-3-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]p(-1,1) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-3-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_4_1" id="input_ps5-3-check_4_1" aria-describedby="status_ps5-3-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_4_1"/>
<span class="status unanswered" id="status_ps5-3-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_4_1" class="answer"/>
<div id="display_ps5-3-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_4_1_dynamath" name="input_ps5-3-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]p(-1,-1) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps5-3-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_5_1" id="input_ps5-3-check_5_1" aria-describedby="status_ps5-3-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_5_1"/>
<span class="status unanswered" id="status_ps5-3-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_5_1" class="answer"/>
<div id="display_ps5-3-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_5_1_dynamath" name="input_ps5-3-check_5_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the covariance and correlation of [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline]. Again, both will be formulas using the parameter <tt class="tt">c</tt>. </p>
<p>
[mathjaxinline]\text {Cov}(X,Y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps5-3-check_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_6_1" id="input_ps5-3-check_6_1" aria-describedby="status_ps5-3-check_6_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_6_1"/>
<span class="status unanswered" id="status_ps5-3-check_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_6_1" class="answer"/>
<div id="display_ps5-3-check_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_6_1_dynamath" name="input_ps5-3-check_6_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]\text {Cor}(X,Y) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_ps5-3-check_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_7_1" id="input_ps5-3-check_7_1" aria-describedby="status_ps5-3-check_7_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_7_1"/>
<span class="status unanswered" id="status_ps5-3-check_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_7_1" class="answer"/>
<div id="display_ps5-3-check_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_7_1_dynamath" name="input_ps5-3-check_7_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">3(b)</b> Your answers will be lists of values of [mathjaxinline]c[/mathjaxinline]. If the list is empty answer 'NONE' (without the quotation marks). </p>
<p>
Give the list of values for which [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] are independent: </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_ps5-3-check_8_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_8_1" id="input_ps5-3-check_8_1" aria-describedby="status_ps5-3-check_8_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_8_1"/>
<span class="status unanswered" id="status_ps5-3-check_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_8_1" class="answer"/>
<div id="display_ps5-3-check_8_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_8_1_dynamath" name="input_ps5-3-check_8_1_dynamath"/>
</div>
</div></div>
</p>
<p>
Give the list of values for which [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] are fully correlated: </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="inputtype_ps5-3-check_9_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps5-3-check_9_1" id="input_ps5-3-check_9_1" aria-describedby="status_ps5-3-check_9_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps5-3-check_9_1"/>
<span class="status unanswered" id="status_ps5-3-check_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-3-check_9_1" class="answer"/>
<div id="display_ps5-3-check_9_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-3-check_9_1_dynamath" name="input_ps5-3-check_9_1_dynamath"/>
</div>
</div></div>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 3" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_ps5-3-check" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ps5-3-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-3-check-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-3-check-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="ps5-3-check-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
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<h3 class="hd hd-3 problem-header" id="ps5-4-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps5-4-check-problem-progress" tabindex="-1">
Problem 4
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<p>
Give all numerical answers to 3 significant digits. </p>
<p><b class="bf">4(a)</b> Give the joint pdf [mathjaxinline]f(a,b)[/mathjaxinline] and cdf [mathjaxinline]F(a,b)[/mathjaxinline]. (Use the variables <tt class="tt">a</tt> and <tt class="tt">b</tt> </p>
<p>
[mathjaxinline]f(a,b) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-4-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4-check_2_1" id="input_ps5-4-check_2_1" aria-describedby="status_ps5-4-check_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_2_1"/>
<span class="status unanswered" id="status_ps5-4-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_2_1" class="answer"/>
<div id="display_ps5-4-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_2_1_dynamath" name="input_ps5-4-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]F(a,b) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-4-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4-check_3_1" id="input_ps5-4-check_3_1" aria-describedby="status_ps5-4-check_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_3_1"/>
<span class="status unanswered" id="status_ps5-4-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_3_1" class="answer"/>
<div id="display_ps5-4-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_3_1_dynamath" name="input_ps5-4-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">4(b)</b> Give the probability Alicia arrives before 12:30. </p>
<p>
[mathjaxinline]P(A \le 30) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps5-4-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps5-4-check_4_1" id="input_ps5-4-check_4_1" aria-describedby="status_ps5-4-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_4_1"/>
<span class="status unanswered" id="status_ps5-4-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_4_1" class="answer"/>
<div id="display_ps5-4-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_4_1_dynamath" name="input_ps5-4-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bf">4(c)</b> Give the probability Alicia arrives before 12:15 and Bernardo arrives between 12:30 and 12:45. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps5-4-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
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<input type="text" name="input_ps5-4-check_5_1" id="input_ps5-4-check_5_1" aria-describedby="status_ps5-4-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_5_1"/>
<span class="status unanswered" id="status_ps5-4-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_5_1" class="answer"/>
<div id="display_ps5-4-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_5_1_dynamath" name="input_ps5-4-check_5_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">4(d)</b> Give the probability Alicia arrives less than 5 minutes after Bernardo. (Note, this includes the possibility that she arrives before him.) </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ps5-4-check_6_1" class="text-input-dynamath capa_inputtype inline textline">
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<input type="text" name="input_ps5-4-check_6_1" id="input_ps5-4-check_6_1" aria-describedby="status_ps5-4-check_6_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_6_1"/>
<span class="status unanswered" id="status_ps5-4-check_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_6_1" class="answer"/>
<div id="display_ps5-4-check_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_6_1_dynamath" name="input_ps5-4-check_6_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">4(e)</b> Give the probability Alicia and Bernardo have lunch together. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_ps5-4-check_7_1" class="text-input-dynamath capa_inputtype inline textline">
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<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
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<input type="text" name="input_ps5-4-check_7_1" id="input_ps5-4-check_7_1" aria-describedby="status_ps5-4-check_7_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps5-4-check_7_1"/>
<span class="status unanswered" id="status_ps5-4-check_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-4-check_7_1" class="answer"/>
<div id="display_ps5-4-check_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-4-check_7_1_dynamath" name="input_ps5-4-check_7_1_dynamath"/>
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Problem 5
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<p><b class="bf">5</b> Give the covariance and correlation of [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline]. You can use the parameter <tt class="tt">n</tt> in your answers </p>
<p>
[mathjaxinline]\text {Cov}(X,Y) =[/mathjaxinline] </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps5-5-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
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<input type="text" name="input_ps5-5-check_2_1" id="input_ps5-5-check_2_1" aria-describedby="status_ps5-5-check_2_1" value="" class="math" size="60"/>
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<textarea style="display:none" id="input_ps5-5-check_2_1_dynamath" name="input_ps5-5-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
[mathjaxinline]\text {Cor}(X,Y) =[/mathjaxinline] </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps5-5-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/MJxPrep.js"/>
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<input type="text" name="input_ps5-5-check_3_1" id="input_ps5-5-check_3_1" aria-describedby="status_ps5-5-check_3_1" value="" class="math" size="60"/>
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<span class="status unanswered" id="status_ps5-5-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps5-5-check_3_1" class="answer"/>
<div id="display_ps5-5-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps5-5-check_3_1_dynamath" name="input_ps5-5-check_3_1_dynamath"/>
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