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<h2 class="hd hd-2 unit-title">Class 4 R reading questions</h2>
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Problem 1
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Using what you've learned about R, simulate rolling one die 100000 times and compute the average of the 100000 rolls. (You should use the sample() and mean() functions.) </p>
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Problem 2
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In a sequence of numbers a run is a subsequence of the same number. For example (1,1,1,2,3,3,3,1,1) has a run of 1's of length 3, a run of 2's of length 1, a run of 3's of length 3 and a run of 1's of length 2. The R function rle() helps compute the run lengths in a sequence. Try running the following code </p>
<p>
<tt class="ttfamily">x = c(1,1,1,2,3,3,3,1,1)<br/>y = rle(x)<br/>print(y) </tt>
</p>
<p>
You should see the following<br/><tt class="ttfamily">Run Length Encoding<br/> lengths: int [1:4] 3 1 3 2<br/> values : num [1:4] 1 2 3 1 </tt> </p>
<p>
The lengths vector shows the runs in <tt class="ttfamily">x</tt> and the values vector shows the number used for each run. To pick out just the lengths vector you use the command: &#8201; y$lengths. </p>
<p>
Give the commannd:<br/><tt class="ttfamily">print(y$lengths)</tt> </p>
<p>
You should see: &#8201; <tt class="ttfamily">[1] 3 1 3 2</tt> </p>
<p>
Run the code<br/><tt class="ttfamily">set.seed(10)<br/>y = rbinom(20,1,.5) </tt> </p>
<p><tt class="ttfamily">y</tt> is a vector of 0's and 1's of length 20. </p>
<p>
Use <tt class="ttfamily">rle(y)</tt> to find the length of the longest run in <tt class="ttfamily">y</tt>. </p>
<p>
Give the length of the longest run.<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_c4-r-rq2_2_1" class=" capa_inputtype textline">
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<h2 class="hd hd-2 unit-title">Discrete random variables</h2>
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<h2 class="hd hd-2 unit-title">Class 4 reading questions: 1</h2>
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Problem 1
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The random variable [mathjaxinline]X[/mathjaxinline] has the following table. </p>
<center>
<table class="tabular" cellspacing="0" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
values [mathjaxinline]a[/mathjaxinline]: </td>
<td style="text-align:center; border:none">
5 </td>
<td style="text-align:center; border:none">
10 </td>
<td style="text-align:center; border:none">
15 </td>
<td style="text-align:center; border:none">
20</td>
</tr>
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pmf [mathjaxinline]p(a)[/mathjaxinline]: </td>
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2/10 </td>
<td style="text-align:center; border:none">
1/10 </td>
<td style="text-align:center; border:none">
2/10 </td>
<td style="text-align:center; border:none">
? </td>
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<p><b class="bf">(a)</b> What is [mathjaxinline]p(20)[/mathjaxinline]? </p>
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Let [mathjaxinline]F(a)[/mathjaxinline] be the cumulative distribution function. </p>
<p><b class="bf">(b)</b> What is [mathjaxinline]F(17)[/mathjaxinline]? </p>
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<p><b class="bf">(c)</b> What is [mathjaxinline]F(20)[/mathjaxinline]? </p>
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<p><b class="bf">(d)</b> What is [mathjaxinline]F(25)[/mathjaxinline]? </p>
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Problem 2
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<p>
Suppose [mathjaxinline]X \sim \text {binomial}(6,0.5)[/mathjaxinline]. What is [mathjaxinline]P(X=3)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">Expectation</h2>
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<p>You can download this pdf file from <a href="/assets/courseware/v1/945558e1113b98a4e32e228a3def19fc/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class4-prep-b.pdf">here</a></p><iframe src="/assets/courseware/v1/945558e1113b98a4e32e228a3def19fc/asset-v1:MITx+18.05r_10+2022_Summer+type@asset+block/pdf_mit18_05_s22_class4-prep-b.pdf" width="100%" height="1000px"> </iframe>
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<h2 class="hd hd-2 unit-title">Class 4 reading questions: 2</h2>
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Problem 1
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<p>
Let [mathjaxinline]X \sim \text {Bernoulli(1/3)}[/mathjaxinline]. </p>
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(a) What is [mathjaxinline]E[X][/mathjaxinline]? </p>
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</p>
<p>
(b) What is [mathjaxinline]E[X^2][/mathjaxinline]? </p>
<p>
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