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<h2 class="hd hd-2 unit-title">Joint distributions: independence</h2>
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<h2 class="hd hd-2 unit-title">Class 7 reading questions: joint distributions</h2>
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<p><b class="bf">1.</b> [mathjaxinline]f(x,y) = cxy[/mathjaxinline] is a joint pdf on [mathjaxinline][0,1]\times [0,1][/mathjaxinline]. What is the value of [mathjaxinline]c[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c7-rq11_2_1" class=" capa_inputtype textline">
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<p><b class="bf">2.</b> Consider the following joint probability table. </p>
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<p><b class="bf">(a)</b> What is the probability that [mathjaxinline]X \le 2[/mathjaxinline] and [mathjaxinline]Y \le 2[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq12_2_1">
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<p><b class="bf">(b)</b> What is the marginal probability for [mathjaxinline]X = 1[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq12_3_1">
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<p><b class="bf">(c)</b> Are [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] independent? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq12_4_1">
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<h2 class="hd hd-2 unit-title">Covariance and correlation</h2>
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<h2 class="hd hd-2 unit-title">Class 7 reading questions: covariance and correlation</h2>
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<p><b class="bf">1.</b> (a) Suppose [mathjaxinline]\text {Cov}(X,Y) \ne 0[/mathjaxinline]. Are [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] independent? </p>
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<text> Yes</text>
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<input type="radio" name="input_c7-rq21_2_1" id="input_c7-rq21_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c7-rq21_2_1-choice_2-label" for="input_c7-rq21_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c7-rq21_2_1">
<text> No</text>
</label>
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<div class="field">
<input type="radio" name="input_c7-rq21_2_1" id="input_c7-rq21_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c7-rq21_2_1-choice_3-label" for="input_c7-rq21_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c7-rq21_2_1">
<text> Maybe</text>
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<span id="answer_c7-rq21_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c7-rq21_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
(b) Suppose [mathjaxinline]\text {Cov}(X,Y) = 0[/mathjaxinline]. Are [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] independent? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq21_3_1">
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<input type="radio" name="input_c7-rq21_3_1" id="input_c7-rq21_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c7-rq21_3_1-choice_1-label" for="input_c7-rq21_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c7-rq21_3_1">
<text> Yes</text>
</label>
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<div class="field">
<input type="radio" name="input_c7-rq21_3_1" id="input_c7-rq21_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c7-rq21_3_1-choice_2-label" for="input_c7-rq21_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c7-rq21_3_1">
<text> No</text>
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<div class="field">
<input type="radio" name="input_c7-rq21_3_1" id="input_c7-rq21_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c7-rq21_3_1-choice_3-label" for="input_c7-rq21_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c7-rq21_3_1">
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<span id="answer_c7-rq21_3_1"/>
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<span class="status unanswered" id="status_c7-rq21_3_1" data-tooltip="Not yet answered.">
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<div class="vert vert-1" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c7-rq22">
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data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c7-rq22" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c7-rq22/handler/xmodule_handler"
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Problem 2
</h3>
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<div class="problem">
<div>
<p><b class="bf">2.</b> Consider the following joint probability table. </p>
<center>
<img src="/assets/courseware/v1/7c1b52e805a67a99f9174fbd45cf2eac/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/images_figc7-16.svg" width="250px" style="margin: 0px 10px 10px 10px"/>
</center>
<p><b class="bf">(a)</b> Are [mathjaxinline]X[/mathjaxinline] and [mathjaxinline]Y[/mathjaxinline] independent? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq22_2_1">
<fieldset aria-describedby="status_c7-rq22_2_1">
<div class="field">
<input type="radio" name="input_c7-rq22_2_1" id="input_c7-rq22_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c7-rq22_2_1-choice_1-label" for="input_c7-rq22_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_2_1">
<text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_2_1" id="input_c7-rq22_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c7-rq22_2_1-choice_2-label" for="input_c7-rq22_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_2_1">
<text> No</text>
</label>
</div>
<span id="answer_c7-rq22_2_1"/>
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<div class="indicator-container">
<span class="status unanswered" id="status_c7-rq22_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bf">(b)</b> What is [mathjaxinline]\text {Cov}(X,Y)[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c7-rq22_3_1">
<fieldset aria-describedby="status_c7-rq22_3_1">
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c7-rq22_3_1-choice_1-label" for="input_c7-rq22_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> -0.3</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c7-rq22_3_1-choice_2-label" for="input_c7-rq22_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> -0.2</text>
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<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c7-rq22_3_1-choice_3-label" for="input_c7-rq22_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> -0.1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="c7-rq22_3_1-choice_4-label" for="input_c7-rq22_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> 0</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="c7-rq22_3_1-choice_5-label" for="input_c7-rq22_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> 0.1</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="c7-rq22_3_1-choice_6-label" for="input_c7-rq22_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> 0.2</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c7-rq22_3_1" id="input_c7-rq22_3_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="c7-rq22_3_1-choice_7-label" for="input_c7-rq22_3_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_c7-rq22_3_1">
<text> 0.3</text>
</label>
</div>
<span id="answer_c7-rq22_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c7-rq22_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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</div></div>
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<span id="solution_c7-rq22_solution_1"/>
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