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<h1 class="heading"><a href="my-great-book.html"><span class="title">Math 1044 Notes</span></a></h1>
<p class="byline">Andy Eisenberg</p>
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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-Set-Theory.html" class="internal"><span class="codenumber">1.1</span> <span class="title">Set 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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<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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<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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<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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<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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<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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<div class="toc-title-box"><a href="sec-Covariance.html" class="internal"><span class="codenumber">3.3</span> <span class="title">Covariance</span></a></div>
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<div class="toc-title-box"><a href="ch-Confidence-Intervals.html" class="internal"><span class="codenumber">4</span> <span class="title">Confidence Intervals</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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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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 active"><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>
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<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>
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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>
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<main class="ptx-main"><div id="ptx-content" class="ptx-content"><section class="section" id="sec-Correlation"><h2 class="heading hide-type">
<span class="type">Section</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6.1</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="title">Correlation</span>
</h2>
<div class="para" id="sec-Correlation-2">Text of section.<div class="autopermalink" data-description="Paragraph"><a href="#sec-Correlation-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<section class="exercises" id="exercises-Correlation"><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-Correlation-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-Correlation-1-1-1">Calculate the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Correlation-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-Correlation-1-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6.1.1<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 6.1.1: "><a href="#exercises-Correlation-1-1-2" title="Copy heading and permalink for Table 6.1.1: " aria-label="Copy heading and permalink for Table 6.1.1: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</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">0.12</td>
<td class="c m b1 r0 l0 t0 lines">0.24</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">34</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 6.1.1"><a href="#exercises-Correlation-1" title="Copy heading and permalink for Exercise 6.1.1" aria-label="Copy heading and permalink for Exercise 6.1.1">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Correlation-2"><h4 class="heading"><span class="codenumber">2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Correlation-2-1-1">Calculate the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{:}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Correlation-2-1-1" 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-Correlation-2-1-2"><figcaption><span class="type">Table</span><span class="space heading-divison-mark heading-divison-mark__space"> </span><span class="codenumber">6.1.2<span class="period heading-divison-mark heading-divison-mark__period">.</span></span><span class="space heading-divison-mark heading-divison-mark__space"> </span><div class="autopermalink" data-description="Table 6.1.2: "><a href="#exercises-Correlation-2-1-2" title="Copy heading and permalink for Table 6.1.2: " aria-label="Copy heading and permalink for Table 6.1.2: ">🔗</a></div></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr class="header-horizontal">
<th scope="col" class="c m b1 r1 l0 t0 lines"></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 0\)</span></th>
<th scope="col" class="c m b1 r1 l0 t0 lines"><span class="process-math">\(X = 1\)</span></th>
<th scope="col" class="c m b1 r0 l0 t0 lines"><span class="process-math">\(X = 2\)</span></th>
</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">0.08</td>
<td class="c m b1 r1 l0 t0 lines">0.16</td>
<td class="c m b1 r0 l0 t0 lines">0.1</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">0.14</td>
<td class="c m b0 r1 l0 t0 lines">0.2</td>
<td class="c m b0 r0 l0 t0 lines">0.32</td>
</tr>
</table></div></figure><div class="autopermalink" data-description="Exercise 6.1.2"><a href="#exercises-Correlation-2" title="Copy heading and permalink for Exercise 6.1.2" aria-label="Copy heading and permalink for Exercise 6.1.2">🔗</a></div></article><article class="exercise exercise-like" id="exercises-Correlation-3"><h4 class="heading"><span class="codenumber">3<span class="period heading-divison-mark heading-divison-mark__period">.</span></span></h4>
<div class="para" id="exercises-Correlation-3-1-1">Suppose we roll a fair, 4-sided die two times. Let <span class="process-math">\(X\)</span> be the sum of the rolls, and let <span class="process-math">\(Y\)</span> be the product of the rolls. Find the correlation of <span class="process-math">\(X\)</span> and <span class="process-math">\(Y\text{.}\)</span><div class="autopermalink" data-description="Paragraph"><a href="#exercises-Correlation-3-1-1" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div> <div class="para" id="exercises-Correlation-3-1-2">[Note: Somewhat tedious.]<div class="autopermalink" data-description="Paragraph"><a href="#exercises-Correlation-3-1-2" title="Copy heading and permalink for Paragraph" aria-label="Copy heading and permalink for Paragraph">🔗</a></div>
</div>
<div class="autopermalink" data-description="Exercise 6.1.3"><a href="#exercises-Correlation-3" title="Copy heading and permalink for Exercise 6.1.3" aria-label="Copy heading and permalink for Exercise 6.1.3">🔗</a></div></article><div class="autopermalink" data-description="Exercises 6.1"><a href="#exercises-Correlation" title="Copy heading and permalink for Exercises 6.1" aria-label="Copy heading and permalink for Exercises 6.1">🔗</a></div></section><div class="autopermalink" data-description="Section 6.1: Correlation"><a href="#sec-Correlation" title="Copy heading and permalink for Section 6.1: Correlation" aria-label="Copy heading and permalink for Section 6.1: Correlation">🔗</a></div></section></div>
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