<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="VerticalStudentView" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@class13-prep-a" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Bayesian updating: continuous priors</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class13-prep-a_hashpdf">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class13-prep-a_hashpdf" data-runtime-class="LmsRuntime" data-has-score="False">
<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/d0ba5aa88ab1826255a244e566d17f26/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class13-prep-a.pdf">here</a></p><iframe src="/assets/courseware/v1/d0ba5aa88ab1826255a244e566d17f26/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class13-prep-a.pdf" width="100%" height="1000px"> </iframe>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="VerticalStudentView" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@class13-prep-b" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Bayesian updating: notational conventions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class13-prep-b_hashpdf">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class13-prep-b_hashpdf" data-runtime-class="LmsRuntime" data-has-score="False">
<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/v1/53b09088021e2ab878a981278a4ce2c1/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class13-prep-b.pdf">here</a></p><iframe src="/assets/courseware/v1/53b09088021e2ab878a981278a4ce2c1/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class13-prep-b.pdf" width="100%" height="1000px"> </iframe>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="VerticalStudentView" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@class13-rq1-vertical" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Class 13 reading questions: Bayesian updating continuous priors</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c13-rq11" class="problems-wrapper" role="group"
aria-labelledby="c13-rq11-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c13-rq11-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11-problem-progress" tabindex="-1">
Problem 1
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq11-problem-progress"></div>
<div class="problem">
<div>
<p>
For these problems the geometric distribution [mathjaxinline]\text {geom}(\theta )[/mathjaxinline] represents the number of successes until the first failure in a sequence of independent Bernoulli trials, where [mathjaxinline]\theta[/mathjaxinline] is the probability of success for each trial. </p>
<p><b class="bf">1.</b> Assume [mathjaxinline]X \sim \text {geometric}(\theta )[/mathjaxinline] where the parameter [mathjaxinline]\theta[/mathjaxinline] lies in the [mathjaxinline][0,1][/mathjaxinline]. We don't know the value of [mathjaxinline]\theta[/mathjaxinline] but have good information that it is most likely to be near 0.5. Which of the following possible priors most closely matches our information? </p>
<center><img src="/assets/courseware/v1/23a8dd091d219d63a3ebef7ffb956f5d/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/images_figc13-rq1-1.svg" width="300px" style="margin: 0px 10px 10px 10px"/> &#8195;<img src="/assets/courseware/v1/a0d874e1ebe1f118beb320a02b8d2f12/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/images_figc13-rq1-2.svg" width="300px" style="margin: 0px 10px 10px 10px"/> &#8195;<img src="/assets/courseware/v1/791c432617555d463ea6fc147d57f9e5/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/images_figc13-rq1-3.svg" width="300px" style="margin: 0px 10px 10px 10px"/> </center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c13-rq11_2_1">
<fieldset aria-describedby="status_c13-rq11_2_1">
<div class="field">
<input type="radio" name="input_c13-rq11_2_1" id="input_c13-rq11_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c13-rq11_2_1-choice_1-label" for="input_c13-rq11_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c13-rq11_2_1">
<text> A</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c13-rq11_2_1" id="input_c13-rq11_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c13-rq11_2_1-choice_2-label" for="input_c13-rq11_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c13-rq11_2_1">
<text> B</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c13-rq11_2_1" id="input_c13-rq11_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c13-rq11_2_1-choice_3-label" for="input_c13-rq11_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c13-rq11_2_1">
<text> C</text>
</label>
</div>
<span id="answer_c13-rq11_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c13-rq11_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>
<div class="solution-span">
<span id="solution_c13-rq11_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 1" />
<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_c13-rq11" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c13-rq11">
<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="c13-rq11-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="c13-rq11-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="c13-rq11-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="True">
<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-1" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c13-rq12" class="problems-wrapper" role="group"
aria-labelledby="c13-rq12-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c13-rq12-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq12-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2.</b> Suppose we take one draw from the random variable [mathjaxinline]X \sim \text {geom}(\theta )[/mathjaxinline] and get the data [mathjaxinline]x=3[/mathjaxinline]. For a fixed hypothesis for the value of [mathjaxinline]\theta[/mathjaxinline] what is the likelihood function [mathjaxinline]p(x|\theta )[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_c13-rq12_2_1">
<fieldset aria-describedby="status_c13-rq12_2_1">
<div class="field">
<input type="radio" name="input_c13-rq12_2_1" id="input_c13-rq12_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="c13-rq12_2_1-choice_1-label" for="input_c13-rq12_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_c13-rq12_2_1">
<text> [mathjaxinline]\theta ^3(1-\theta )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c13-rq12_2_1" id="input_c13-rq12_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c13-rq12_2_1-choice_2-label" for="input_c13-rq12_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c13-rq12_2_1">
<text> [mathjaxinline]\binom {3}{1}\theta ^3(1-\theta )[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_c13-rq12_2_1" id="input_c13-rq12_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="c13-rq12_2_1-choice_3-label" for="input_c13-rq12_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_c13-rq12_2_1">
<text> [mathjaxinline]p^3(1-p)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_c13-rq12_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_c13-rq12_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>
<div class="solution-span">
<span id="solution_c13-rq12_solution_1"/>
</div></p>
</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_c13-rq12" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c13-rq12">
<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="c13-rq12-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="c13-rq12-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="c13-rq12-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="True">
<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@c13-rq13">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq13" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c13-rq13" class="problems-wrapper" role="group"
aria-labelledby="c13-rq13-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq13" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq13/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c13-rq13-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq13-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq13-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">3.</b> Again, the data is drawn from a [mathjaxinline]\text {geom}(\theta )[/mathjaxinline] distribution with unknown parameter [mathjaxinline]\theta[/mathjaxinline]. Assume your prior probability density for [mathjaxinline]\theta[/mathjaxinline] is [mathjaxinline]f(\theta ) = 2\theta[/mathjaxinline]. What is the prior predictive probability that [mathjaxinline]x = 3[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c13-rq13_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<input type="text" name="input_c13-rq13_2_1" id="input_c13-rq13_2_1" aria-describedby="status_c13-rq13_2_1" value=""/>
<span class="trailing_text" id="trailing_text_c13-rq13_2_1"/>
<span class="status unanswered" id="status_c13-rq13_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c13-rq13_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_c13-rq13_solution_1"/>
</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_c13-rq13" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c13-rq13">
<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="c13-rq13-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="c13-rq13-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="c13-rq13-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="True">
<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@c13-rq14">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq14" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_c13-rq14" class="problems-wrapper" role="group"
aria-labelledby="c13-rq14-problem-title"
data-problem-id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq14" data-url="/courses/course-v1:MITx+18.05r_10+2022_Summer/xblock/block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq14/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="c13-rq14-problem-title" aria-describedby="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq14-problem-progress" tabindex="-1">
Problem 4
</h3>
<div class="problem-progress" id="block-v1:MITx+18.05r_10+2022_Summer+type@problem+block@c13-rq14-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4.</b> Our data is one draw from a [mathjaxinline]\text {geometric}(\theta )[/mathjaxinline] distribution with unknown parameter [mathjaxinline]\theta[/mathjaxinline]. Assume your prior probability density for [mathjaxinline]\theta[/mathjaxinline] is [mathjaxinline]f(\theta ) = 2\theta[/mathjaxinline]. Also suppose the data [mathjaxinline]x = 3[/mathjaxinline]. Give the formula for the posterior pdf of [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
(You should enter the formula using R notation, e.g. <tt class="ttfamily">theta*(2+theta^2)</tt>) </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_c13-rq14_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_c13-rq14_2_1" id="input_c13-rq14_2_1" data-input-id="c13-rq14_2_1" value="" aria-describedby="status_c13-rq14_2_1" size="50"/>
<span class="trailing_text" id="trailing_text_c13-rq14_2_1"/>
<span class="status unanswered" id="status_c13-rq14_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_c13-rq14_2_1" class="answer"/>
<div id="input_c13-rq14_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_c13-rq14_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 4" />
<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_c13-rq14" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_c13-rq14">
<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="c13-rq14-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="c13-rq14-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="c13-rq14-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="True">
<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>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="VerticalStudentView" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@vertical+block@class14-prep-b" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">(Optional reading) Updating: continuous priors and data</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class14-prep-b_hashpdf">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+18.05r_10+2022_Summer" data-graded="True" data-init="XBlockToXModuleShim" data-request-token="cd601e781edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+18.05r_10+2022_Summer+type@html+block@class14-prep-b_hashpdf" data-runtime-class="LmsRuntime" data-has-score="False">
<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/v1/8012b33e5920cda0f41cd5f5d277e6ce/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class14-prep-b.pdf">here</a></p><iframe src="/assets/courseware/v1/8012b33e5920cda0f41cd5f5d277e6ce/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class14-prep-b.pdf" width="100%" height="1000px"> </iframe>
</div>
</div>
</div>
</div>