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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
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<div class="toc-title-box"><a href="ch-Introduction.html" class="internal"><span class="codenumber">0</span> <span class="title">Introduction</span></a></div>
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<div class="toc-title-box"><a href="ch-Probability.html" class="internal"><span class="codenumber">1</span> <span class="title">Probability Theory</span></a></div>
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<div class="toc-title-box"><a href="sec-Probability.html" class="internal"><span class="codenumber">1.2</span> <span class="title">Definition of Probability</span></a></div>
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<div class="toc-title-box"><a href="sec-Conditional-Probability.html" class="internal"><span class="codenumber">1.3</span> <span class="title">Conditional Probability</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-conditional-probability" class="internal"><span class="codenumber">1.3.1</span> <span class="title">Conditional Probability</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#subsec-diagnostic-testing" class="internal"><span class="codenumber">1.3.2</span> <span class="title">Diagnostic Testing</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Conditional-Probability.html#exercises-Conditional-Probability" class="internal"><span class="codenumber">1.3.3</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Independent-Events.html" class="internal"><span class="codenumber">1.4</span> <span class="title">Independent Events</span></a></div>
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<div class="toc-title-box"><a href="ch-Random-Variables.html" class="internal"><span class="codenumber">2</span> <span class="title">Random Variables</span></a></div>
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<div class="toc-title-box"><a href="sec-Discrete-RVs.html" class="internal"><span class="codenumber">2.1</span> <span class="title">Discrete Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-Discrete-RVs" class="internal"><span class="codenumber">2.1.1</span> <span class="title">Discrete Random Variables</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-binomial-distribution" class="internal"><span class="codenumber">2.1.2</span> <span class="title">The Binomial Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-geometric-distribution" class="internal"><span class="codenumber">2.1.3</span> <span class="title">Geometric Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#subsec-poisson-distribution" class="internal"><span class="codenumber">2.1.4</span> <span class="title">Poisson Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Discrete-RVs.html#exercises-Discrete-RVs" class="internal"><span class="codenumber">2.1.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Continuous-RVs.html" class="internal"><span class="codenumber">2.2</span> <span class="title">Continuous Random Variables</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-pdfs" class="internal"><span class="codenumber">2.2.1</span> <span class="title">Probability Density Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-cdf" class="internal"><span class="codenumber">2.2.2</span> <span class="title">Cumulative Distribution Functions</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-exponential-distribution" class="internal"><span class="codenumber">2.2.3</span> <span class="title">The Exponential Distribution</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#subsec-normal-distribution" class="internal"><span class="codenumber">2.2.4</span> <span class="title">Normal Distribution</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Continuous-RVs.html#exercises-Continuous-RVs" class="internal"><span class="codenumber">2.2.5</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Joint-Distributions.html" class="internal"><span class="codenumber">2.3</span> <span class="title">Joint Distributions</span></a></div>
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<ul class="structural toc-item-list active"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Joint-Distributions.html#exercises-Joint-Distributions" class="internal"><span class="codenumber">2.3</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="ch-Expected-Value.html" class="internal"><span class="codenumber">3</span> <span class="title">Expected Value and Variance</span></a></div>
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<div class="toc-title-box"><a href="sec-Expected-Value.html" class="internal"><span class="codenumber">3.1</span> <span class="title">Expected Value</span></a></div>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-discrete-EV" class="internal"><span class="codenumber">3.1.1</span> <span class="title">Discrete Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-continuous-EV" class="internal"><span class="codenumber">3.1.2</span> <span class="title">Continuous Expected Value</span></a></div></li>
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<li class="toc-item toc-subsection"><div class="toc-title-box"><a href="sec-Expected-Value.html#subsec-linearity-EV" class="internal"><span class="codenumber">3.1.3</span> <span class="title">Linearity of Expected Value</span></a></div></li>
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<li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Expected-Value.html#exercises-Expected-Value" class="internal"><span class="codenumber">3.1.4</span> <span class="title">Exercises</span></a></div></li>
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<div class="toc-title-box"><a href="sec-Variance.html" class="internal"><span class="codenumber">3.2</span> <span class="title">Variance</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Variance.html#exercises-Variance" class="internal"><span class="codenumber">3.2</span> <span class="title">Exercises</span></a></div></li></ul>
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<div class="toc-title-box"><a href="sec-Likelihood.html" class="internal"><span class="codenumber">4.1</span> <span class="title">Likelihood</span></a></div>
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<div class="toc-title-box"><a href="sec-CLT.html" class="internal"><span class="codenumber">4.2</span> <span class="title">Central Limit Theorem</span></a></div>
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<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-CLT.html#exercises-CLT" class="internal"><span class="codenumber">4.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Confidence-Intervals.html" class="internal"><span class="codenumber">4.3</span> <span class="title">Confidence Intervals</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Confidence-Intervals.html#exercises-Confidence-Intervals" class="internal"><span class="codenumber">4.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Hypothesis-Testing.html" class="internal"><span class="codenumber">5</span> <span class="title">Hypothesis Testing</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-One-Sample-Tests.html" class="internal"><span class="codenumber">5.1</span> <span class="title">One Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-One-Sample-Tests.html#exercises-One-Sample-Tests" class="internal"><span class="codenumber">5.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Two-Sample-Tests.html" class="internal"><span class="codenumber">5.2</span> <span class="title">Two Sample Tests</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Two-Sample-Tests.html#exercises-Two-Sample-Tests" class="internal"><span class="codenumber">5.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Power.html" class="internal"><span class="codenumber">5.3</span> <span class="title">Power of a Test</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Power.html#exercises-Power" class="internal"><span class="codenumber">5.3</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Chi-Squared.html" class="internal"><span class="codenumber">5.4</span> <span class="title"><span class="process-math">\(\chi^2\)</span> Test</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Chi-Squared.html#exercises-Chi-Squared" class="internal"><span class="codenumber">5.4</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
</ul>
|
||
</li>
|
||
<li class="toc-item toc-chapter">
|
||
<div class="toc-title-box"><a href="ch-Linear-Regression.html" class="internal"><span class="codenumber">6</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list">
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Correlation.html" class="internal"><span class="codenumber">6.1</span> <span class="title">Correlation</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Correlation.html#exercises-Correlation" class="internal"><span class="codenumber">6.1</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
</li>
|
||
<li class="toc-item toc-section">
|
||
<div class="toc-title-box"><a href="sec-Linear-Regression.html" class="internal"><span class="codenumber">6.2</span> <span class="title">Linear Regression</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-exercises"><div class="toc-title-box"><a href="sec-Linear-Regression.html#exercises-Linear-Regression" class="internal"><span class="codenumber">6.2</span> <span class="title">Exercises</span></a></div></li></ul>
|
||
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|
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|
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<div class="toc-title-box"><a href="backmatter.html" class="internal"><span class="title">Backmatter</span></a></div>
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<li class="toc-item toc-appendix">
|
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<div class="toc-title-box"><a href="backmatter-2.html" class="internal"><span class="codenumber">A</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table</span></a></div>
|
||
<ul class="structural toc-item-list"><li class="toc-item toc-section"><div class="toc-title-box"><a href="app-Phi-table.html" class="internal"><span class="codenumber">A.1</span> <span class="title"><span class="process-math">\(\Phi(z)\)</span> Table of Values</span></a></div></li></ul>
|
||
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|
||
<li class="toc-item toc-colophon"><div class="toc-title-box"><a href="backmatter-3.html" class="internal"><span class="title">Colophon</span></a></div></li>
|
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|
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Joint-Distributions"><h2 class="heading hide-type">
|
||
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Joint Distributions</span>
|
||
</h2>
|
||
<div class="para" id="sec-Joint-Distributions-2">Sometimes multiple measurements are taken simultaneously and data is naturally grouped by combinations of measurement values. Consider a poll that asks respondents two yes or no questions. It may be natural to report counts of respondents who answered yes/yes, yes/no, no/yes, no/no.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Joint-Distributions-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-joint-distribution"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.1</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="def-joint-distribution-1-1">Let <span class="process-math">\(X\)</span> be a random variable taking values <span class="process-math">\(x_1, x_2, \dotsc, x_m\text{.}\)</span> Let <span class="process-math">\(Y\)</span> be a random variable taking values <span class="process-math">\(y_1, y_2, \dotsc, y_n\text{.}\)</span> The <dfn class="terminology">joint distribution</dfn> for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is the collection of probabilities <span class="process-math">\(\Pr(X = x_i, Y = y_j)\)</span> for all <span class="process-math">\(1 \leq i \leq m, 1 \leq j \leq n\text{.}\)</span> The individual distributions for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> separately are called the <dfn class="terminology">marginal distributions</dfn>.<div class="autopermalink" data-description="Paragraph"><a href="#def-joint-distribution-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 2.3.1"><a href="#def-joint-distribution" title="Copy heading and permalink for Definition 2.3.1" aria-label="Copy heading and permalink for Definition 2.3.1">🔗</a></div></article><div class="para" id="sec-Joint-Distributions-4">In the probability <span class="process-math">\(\Pr(X = x_i, Y = y_j)\text{,}\)</span> the comma should be read as "and". Think of the conditions <span class="process-math">\(X = x_i\)</span> and <span class="process-math">\(Y = y_j\)</span> as events, and <span class="process-math">\(\Pr(X = x_i, Y = y_j)\)</span> is the probability of the intersection of these two events.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Joint-Distributions-4" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="para" id="sec-Joint-Distributions-5">We’re restricting our attention here to joint distributions of two random variables, each taking only finitely many values. It shouldn’t be hard to imagine how to generalize this definition for a joint distribution of three or more discrete random variables. A bit more adjustment would need to be made for a joint distribution of continuous random variables, or a joint distribution between a discrete and continuous random variable.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Joint-Distributions-5" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="example example-like" id="example-joint-indicators"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.2</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="example-joint-indicators-1-1">A poll asks two yes/no questions. Let <span class="process-math">\(X\)</span> indicate a yes on Question 1 and <span class="process-math">\(Y\)</span> indicate a yes on Question 2. When the data is collected, the following joint distribution for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> is created:<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="example-joint-indicators-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution for Indicator Random Variables<div class="autopermalink" data-description="Table 2.3.3: Joint Distribution for Indicator Random Variables"><a href="#example-joint-indicators-1-2" title="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables" aria-label="Copy heading and permalink for Table 2.3.3: Joint Distribution for Indicator Random Variables">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
|
||
<td class="c m b1 r1 l0 t0 lines">0.1</td>
|
||
<td class="c m b1 r0 l0 t0 lines">0.2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
|
||
<td class="c m b0 r1 l0 t0 lines">0.3</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.4</td>
|
||
</tr>
|
||
</table></div></figure> <div class="para logical" id="example-joint-indicators-1-3">
|
||
<div class="para">Note that the sum of all values in the table is 1; this is a probability distribution, and total probability must be 1. This table doesn’t show the distributions for <span class="process-math">\(X\)</span> or <span class="process-math">\(Y\)</span> individually. The data for the two distributions is mixed together. We can, if we wish, take this information and determine separate distributions for <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span> For the distribution for <span class="process-math">\(X\text{,}\)</span> we need to know <span class="process-math">\(\Pr(X = 0)\)</span> and <span class="process-math">\(\Pr(X = 1)\)</span> (with no reference to <span class="process-math">\(Y\)</span>). We can get these probabilities by summing along the columns of the table:</div>
|
||
<div class="displaymath process-math" id="example-joint-indicators-1-3-9">
|
||
\begin{align*}
|
||
\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\
|
||
\Pr(X = 1) \amp = 0.2 + 0.4 = 0.6
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">To get the distribution for <span class="process-math">\(Y\text{,}\)</span> we should add along the rows:</div>
|
||
<div class="displaymath process-math" id="example-joint-indicators-1-3-11">
|
||
\begin{align*}
|
||
\Pr(Y = 0) \amp = 0.1 + 0.2 = 0.3 \\
|
||
\Pr(Y = 1) \amp = 0.3 + 0.4 = 0.7
|
||
\end{align*}
|
||
</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-joint-indicators-1-3" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 2.3.2"><a href="#example-joint-indicators" title="Copy heading and permalink for Example 2.3.2" aria-label="Copy heading and permalink for Example 2.3.2">🔗</a></div></article><div class="para" id="sec-Joint-Distributions-7">If we start with the joint distribution, we can get the marginal distributions. If we start with the marginal distributions, we can’t necessarily find the joint distribution, because we won’t know how the random variables interact.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Joint-Distributions-7" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<article class="definition definition-like" id="def-independent-RVs"><h3 class="heading">
|
||
<span class="type">Definition</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.4</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para" id="def-independent-RVs-1-1">Random variables <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are called <dfn class="terminology">independent</dfn> if the events <span class="process-math">\(X = x\)</span> and <span class="process-math">\(Y = y\)</span> are independent for every possible combination of <span class="process-math">\(x\)</span> and <span class="process-math">\(y\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#def-independent-RVs-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Definition 2.3.4"><a href="#def-independent-RVs" title="Copy heading and permalink for Definition 2.3.4" aria-label="Copy heading and permalink for Definition 2.3.4">🔗</a></div></article><article class="example example-like" id="example-independent-indicators"><h3 class="heading">
|
||
<span class="type">Example</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.5</span><span class="period heading-divison-mark heading-divison-mark__period">.</span>
|
||
</h3>
|
||
<div class="para logical" id="example-independent-indicators-1-1">
|
||
<div class="para">Using the joint distribution and marginal distributions in <a href="sec-Joint-Distributions.html#example-joint-indicators" class="xref" data-knowl="./knowl/xref/example-joint-indicators.html" data-reveal-label="Reveal" data-close-label="Close" title="Example 2.3.2">Example 2.3.2</a>, we can check independence of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> by going cell by cell through the joint distribution table. The top-left cell shows <span class="process-math">\(\Pr(X = 0, Y = 0) = 0.1\text{.}\)</span> The marginal distributions have <span class="process-math">\(\Pr(X = 0) = 0.4\)</span> and <span class="process-math">\(\Pr(Y = 0) = 0.3\text{.}\)</span> If <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> were independent, we would have</div>
|
||
<div class="displaymath process-math" data-contains-math-knowls="./knowl/xref/example-joint-indicators.html" id="example-independent-indicators-1-1-9">
|
||
\begin{align*}
|
||
\Pr(X = 0, Y = 0) \amp = \Pr(X = 0) \Pr(Y = 0) \\
|
||
\text{however, } 0.1 \amp \neq (0.4)(0.3)
|
||
\end{align*}
|
||
</div>
|
||
<div class="para">Therefore, <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\)</span> are not independent.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#example-independent-indicators-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <div class="para" id="example-independent-indicators-1-2">Note that we can stop checking now. As soon as we find one cell in the joint distribution table where the independence condition fails, the two random variables are not independent. In order to establish independence of random variables, <em class="emphasis">all</em> cells in the joint distribution table would have to pass our check.<div class="autopermalink" data-description="Paragraph"><a href="#example-independent-indicators-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Example 2.3.5"><a href="#example-independent-indicators" title="Copy heading and permalink for Example 2.3.5" aria-label="Copy heading and permalink for Example 2.3.5">🔗</a></div></article><section class="exercises" id="exercises-Joint-Distributions"><h3 class="heading hide-type">
|
||
<span class="type">Exercises</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber"></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Exercises</span>
|
||
</h3>
|
||
<article class="exercise exercise-like" id="exercises-Joint-Distributions-1"><h4 class="heading"><span class="codenumber">1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para" id="exercises-Joint-Distributions-1-1-1">Consider <span class="process-math">\(X, Y\)</span> with the joint distribution table below. Are <span class="process-math">\(X, Y\)</span> independent?<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-1-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div> <figure class="table table-like" id="exercises-Joint-Distributions-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.6<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint distribution for <span class="process-math">\(X, Y\)</span><div class="autopermalink" data-description="Table 2.3.6: Joint distribution for X, Y"><a href="#exercises-Joint-Distributions-1-1-2" title="Copy heading and permalink for Table 2.3.6: Joint distribution for X, Y" aria-label="Copy heading and permalink for Table 2.3.6: Joint distribution for X, Y">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
|
||
<td class="c m b1 r1 l0 t0 lines">0.2</td>
|
||
<td class="c m b1 r0 l0 t0 lines">0.3</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
|
||
<td class="c m b0 r1 l0 t0 lines">0.4</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.1</td>
|
||
</tr>
|
||
</table></div></figure><div class="solutions"><details id="exercises-Joint-Distributions-1-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<div class="para" id="exercises-Joint-Distributions-1-2-1">No. For example, <span class="process-math">\(\Pr(X = 0, Y = 0) \neq \Pr(X = 0)\Pr(Y = 0).\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-1-2-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
|
||
</div>
|
||
<div class="autopermalink" data-description="Answer 2.3.1.1"><a href="#exercises-Joint-Distributions-1-2" title="Copy heading and permalink for Answer 2.3.1.1" aria-label="Copy heading and permalink for Answer 2.3.1.1">🔗</a></div>
|
||
</div></details></div>
|
||
<div class="autopermalink" data-description="Exercise 2.3.1"><a href="#exercises-Joint-Distributions-1" title="Copy heading and permalink for Exercise 2.3.1" aria-label="Copy heading and permalink for Exercise 2.3.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Joint-Distributions-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
|
||
<div class="para logical" id="exercises-Joint-Distributions-2-1-1">
|
||
<div class="para">Suppose <span class="process-math">\(X, Y\)</span> have the distributions:</div>
|
||
<div class="displaymath process-math" id="exercises-Joint-Distributions-2-1-1-2">
|
||
\begin{align*}
|
||
\Pr(X = 0) \amp = 0.1 \amp \Pr(Y = 0) \amp = 0.4 \\
|
||
\Pr(X = 1) \amp = 0.4 \amp \Pr(Y = 1) \amp = 0.6 \\
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||
\Pr(X = 2) \amp = 0.5
|
||
\end{align*}
|
||
</div>
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||
<div class="para">Assuming <span class="process-math">\(X, Y\)</span> are independent, write a joint distribution table.</div>
|
||
<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Joint-Distributions-2-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
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</div>
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<div class="solutions"><details id="exercises-Joint-Distributions-2-2" class="answer solution-like born-hidden-knowl"><summary class="knowl__link"><span class="type">Answer</span><span class="period heading-divison-mark heading-divison-mark__period">.</span></summary><div class="answer solution-like knowl__content">
|
||
<figure class="table table-like" id="exercises-Joint-Distributions-2-2-1"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">2.3.7<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span>Joint Distribution<div class="autopermalink" data-description="Table 2.3.7: Joint Distribution"><a href="#exercises-Joint-Distributions-2-2-1" title="Copy heading and permalink for Table 2.3.7: Joint Distribution" aria-label="Copy heading and permalink for Table 2.3.7: Joint Distribution">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"></td>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></td>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></td>
|
||
<td class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b1 r1 l0 t0 lines"><span class="process-math">\(Y = 0\)</span></td>
|
||
<td class="c m b1 r1 l0 t0 lines">0.04</td>
|
||
<td class="c m b1 r1 l0 t0 lines">0.16</td>
|
||
<td class="c m b1 r0 l0 t0 lines">0.2</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="c m b0 r1 l0 t0 lines"><span class="process-math">\(Y = 1\)</span></td>
|
||
<td class="c m b0 r1 l0 t0 lines">0.06</td>
|
||
<td class="c m b0 r1 l0 t0 lines">0.24</td>
|
||
<td class="c m b0 r0 l0 t0 lines">0.3</td>
|
||
</tr>
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||
</table></div></figure><div class="autopermalink" data-description="Answer 2.3.2.1"><a href="#exercises-Joint-Distributions-2-2" title="Copy heading and permalink for Answer 2.3.2.1" aria-label="Copy heading and permalink for Answer 2.3.2.1">🔗</a></div>
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</div></details></div>
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<div class="autopermalink" data-description="Exercise 2.3.2"><a href="#exercises-Joint-Distributions-2" title="Copy heading and permalink for Exercise 2.3.2" aria-label="Copy heading and permalink for Exercise 2.3.2">🔗</a></div></article><div class="autopermalink" data-description="Exercises 2.3"><a href="#exercises-Joint-Distributions" title="Copy heading and permalink for Exercises 2.3" aria-label="Copy heading and permalink for Exercises 2.3">🔗</a></div></section><div class="autopermalink" data-description="Section 2.3: Joint Distributions"><a href="#sec-Joint-Distributions" title="Copy heading and permalink for Section 2.3: Joint Distributions" aria-label="Copy heading and permalink for Section 2.3: Joint Distributions">🔗</a></div></section></div>
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