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<h2 class="hd hd-2 unit-title">Problem set 6</h2>
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<p>You can download this pdf file from <a href="/assets/courseware/v1/c671abb61d3676d2eb5d9cefdf3dc372/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps6.pdf">here</a></p><iframe src="/assets/courseware/v1/c671abb61d3676d2eb5d9cefdf3dc372/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_ps6.pdf" width="100%" height="1000px"> </iframe>
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<h2 class="hd hd-2 unit-title">Pset 6 problem checker</h2>
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data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="ps6-1-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-1-check-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-1-check-problem-progress"></div>
<div class="problem">
<div>
<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</b> Let the product and sum of the data be [mathjaxinline]P = \prod _{i=1}^ m x_ i[/mathjaxinline] and [mathjaxinline]S = \sum _{i=1}^ m x_ i[/mathjaxinline]. Give the MLE for [mathjaxinline]a[/mathjaxinline] in terms of [mathjaxinline]S[/mathjaxinline], [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]m[/mathjaxinline]. </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-1-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_ps6-1-check_2_1" id="input_ps6-1-check_2_1" aria-describedby="status_ps6-1-check_2_1" value="" class="math" size="70"/>
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<span class="status unanswered" id="status_ps6-1-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-1-check_2_1" class="answer"/>
<div id="display_ps6-1-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-1-check_2_1_dynamath" name="input_ps6-1-check_2_1_dynamath"/>
</div>
</div></div>
</p>
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<input type="hidden" name="problem_id" value="Problem 1" />
<div class="submit-attempt-container">
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<div class="submission-feedback" id="submission_feedback_ps6-1-check">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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aria-labelledby="ps6-2a-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2a-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2a-check/handler/xmodule_handler"
data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="ps6-2a-check-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2a-check-problem-progress" tabindex="-1">
Problem 2a
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2a-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2(a)</b> Use the variables and parameters <tt class="tt">y_i, x_i, a, b, sigma</tt>. </p>
<p>
Give the formula for the likelihood of [mathjaxinline]y_ i[/mathjaxinline]. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-2a-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_ps6-2a-check_2_1" id="input_ps6-2a-check_2_1" aria-describedby="status_ps6-2a-check_2_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps6-2a-check_2_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2a-check_2_1" class="answer"/>
<div id="display_ps6-2a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2a-check_2_1_dynamath" name="input_ps6-2a-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the formula for the log likelihood of [mathjaxinline]y_ i[/mathjaxinline]. <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-2a-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_ps6-2a-check_3_1" id="input_ps6-2a-check_3_1" aria-describedby="status_ps6-2a-check_3_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps6-2a-check_3_1"/>
<span class="status unanswered" id="status_ps6-2a-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2a-check_3_1" class="answer"/>
<div id="display_ps6-2a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2a-check_3_1_dynamath" name="input_ps6-2a-check_3_1_dynamath"/>
</div>
</div></div> </p>
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aria-labelledby="ps6-2bc-check-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2bc-check" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2bc-check/handler/xmodule_handler"
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Problem 2bc
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@ps6-2bc-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2(b)</b> Give the values of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] to at least 3 significant digits. </p>
<p>
[mathjaxinline]a =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-2bc-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_ps6-2bc-check_2_1" id="input_ps6-2bc-check_2_1" aria-describedby="status_ps6-2bc-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-2bc-check_2_1"/>
<span class="status unanswered" id="status_ps6-2bc-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2bc-check_2_1" class="answer"/>
<div id="display_ps6-2bc-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2bc-check_2_1_dynamath" name="input_ps6-2bc-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
[mathjaxinline]b =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-2bc-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_ps6-2bc-check_3_1" id="input_ps6-2bc-check_3_1" aria-describedby="status_ps6-2bc-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-2bc-check_3_1"/>
<span class="status unanswered" id="status_ps6-2bc-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-2bc-check_3_1" class="answer"/>
<div id="display_ps6-2bc-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-2bc-check_3_1_dynamath" name="input_ps6-2bc-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
<b class="bf">2(c)</b>
<em>No answer checker given.</em>
</p>
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<input type="hidden" name="problem_id" value="Problem 2bc" />
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<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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Problem 3
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<p><b class="bf">3(a)</b> Give the MLE of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p>
<p>
[mathjaxinline]a =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps6-3-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<div id="display_ps6-3-check_2_1" class="equation">`{::}`</div>
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</div></div> </p>
<p>
[mathjaxinline]b =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps6-3-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"/>
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<input type="text" name="input_ps6-3-check_3_1" id="input_ps6-3-check_3_1" aria-describedby="status_ps6-3-check_3_1" value="" class="math" size="50"/>
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<span class="status unanswered" id="status_ps6-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_ps6-3-check_3_1" class="answer"/>
<div id="display_ps6-3-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-3-check_3_1_dynamath" name="input_ps6-3-check_3_1_dynamath"/>
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</div></div> </p>
<p><b class="bf">3(b)</b> On your paper, you will need to give the answer in terms of [mathjaxinline]x_1, \ldots x_ n[/mathjaxinline]. To confirm your answer, you should be able to answer the following question quickly with no computation. </p>
<p>
If the data in sorted order is </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]12.3, \, 45.6, \, 78.9, \, 102.3, \, 456.7, \, 8910.1[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
Give the MLE of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline]. </p>
<p>
[mathjaxinline]a =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps6-3-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_ps6-3-check_4_1" id="input_ps6-3-check_4_1" aria-describedby="status_ps6-3-check_4_1" value="" class="math" size="50"/>
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<span class="status unanswered" id="status_ps6-3-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_ps6-3-check_4_1" class="answer"/>
<div id="display_ps6-3-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-3-check_4_1_dynamath" name="input_ps6-3-check_4_1_dynamath"/>
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</div></div> </p>
<p>
[mathjaxinline]b =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps6-3-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_ps6-3-check_5_1" id="input_ps6-3-check_5_1" aria-describedby="status_ps6-3-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps6-3-check_5_1"/>
<span class="status unanswered" id="status_ps6-3-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps6-3-check_5_1" class="answer"/>
<div id="display_ps6-3-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps6-3-check_5_1_dynamath" name="input_ps6-3-check_5_1_dynamath"/>
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</div></div> </p>
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