First pass at Chapter 3
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<introduction>
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<p>
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Testing in codespace
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Testing locally
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</p>
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</introduction>
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<!-- Need a joint distribution section first? --> <section xml:id="sec-Covariance" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title>Covariance</title>
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<definition xml:id="def-covariance">
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<statement>
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<p>
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Let <m>X</m> and <m>Y</m> be random variables.
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The <term>covariance</term> of <m>X</m> and <m>Y</m> is:
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<md>
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<mrow> \Cov(X, Y) = \E\left[\left(X - \E(X)\right) \left(Y - \E(Y)\right)\right]. </mrow>
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</md>
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</p>
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</statement>
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</definition>
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<p>
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Text of section.
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Let's write <m>\mu_X = \E(X)</m> and <m>\mu_Y = \E(Y)</m> to make this is a bit cleaner to look at:
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<md>
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<mrow> \Cov(X, Y) = \E\left[\left(X - \mu_X \right) \left(Y - \mu_Y\right)\right]. </mrow>
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</md>
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This looks like a pretty natural generalization of the variance.
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In fact:
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<md>
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<mrow> \Cov(X, X) \amp = \E\left[ (X - \mu_X)(X - \mu_X)\right] = \E\left[\left(X - \mu_X\right)^2\right] = \Var(X). </mrow>
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</md>
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One difference worth noting is that <m>\Var(X)</m>, as the average of squared values, must always be nonnegative.
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The expression <m>(X - \mu_X)(X - \mu_X)</m> can be negative for combinations of <m>X</m> and <m>Y</m> values where one is greater than its average and the other is smaller.
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If this is the average behavior, then <m>\Cov(X, Y)</m> can be negative overall.
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This is useful information: a negative <m>\Cov(X, Y)</m> would imply that (on average, not necessarily always) when <m>X</m> is larger than typical, <m>Y</m> is smaller than typical, and vice versa.
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</p>
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<p>
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Also as with variance, the covariance definition formula gives us some useful insight, but it can be unwieldy to work with.
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There is an alternative formula, analogous to <xref ref="thm-variance-formula"/>:
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</p>
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<theorem xml:id="thm-covariance-formula">
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<statement>
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<p>
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Let <m>X</m> and <m>Y</m> be random variables.
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Then:
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<md>
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<mrow>\Cov(X, Y) = \E\left(XY\right) - \E(X)\E(Y).</mrow>
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</md>
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</p>
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</statement>
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</theorem>
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<p>
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To make use of this formula, it's helpful to have both a joint distribution and marginal distributions.
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The marginal distributions let you find <m>\E(X)</m> and <m>\E(Y)</m>.
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The joint distribution lets you find <m>\E(XY)</m>.
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</p>
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<example>
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<statement>
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<p>
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Let <m>X</m> and <m>Y</m> be indicator random variables with joint and marginal distributions below.
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</p>
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<sidebyside>
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<table>
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<title></title>
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<tabular halign="center">
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<row bottom="minor">
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<cell right="minor"></cell>
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<cell right="minor"><m>X = 0</m></cell>
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<cell><m>X = 1</m></cell>
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</row>
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<row bottom="minor">
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<cell right="minor"><m>Y = 0</m></cell>
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<cell right="minor">0.2</cell>
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<cell>0.1</cell>
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</row>
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<row>
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<cell right="minor"><m>Y = 1</m></cell>
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<cell right="minor">0.2</cell>
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<cell>0.5</cell>
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</row>
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</tabular>
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</table>
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<table>
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<title></title>
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<tabular halign="center">
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<row bottom="minor">
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<cell><m>x</m></cell><cell><m>\Pr(X = x)</m></cell>
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</row>
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<row>
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<cell>0</cell><cell>0.4</cell>
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</row>
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<row>
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<cell>1</cell><cell>0.6</cell>
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</row>
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</tabular>
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</table>
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<table>
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<title></title>
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<tabular halign="center">
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<row bottom="minor">
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<cell><m>y</m></cell><cell><m>\Pr(Y = y)</m></cell>
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</row>
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<row>
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<cell>0</cell><cell>0.3</cell>
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</row>
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<row>
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<cell>1</cell><cell>0.7</cell>
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</row>
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</tabular>
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</table>
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</sidebyside>
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<p>
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Since they're indicator random variables, we can immediately see that <m>\E(X) = 0.6</m> and <m>\E(Y) = 0.7</m> (see <xref ref="example-indicator-EV"/>).
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For <m>\E(XY)</m>, we'll traverse cell by cell through the joint distribution table, constructing products of the form <m>X</m>-value times <m>Y</m>-value times probability:
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<md>
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<mrow> \E(XY) =\, \amp (0)(0)(0.2) + (1)(0)(0.1) </mrow>
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<mrow> \amp (0)(1)(0.2) + (1)(1)(0.5) </mrow>
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<mrow> =\, \amp 0.5. </mrow>
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</md>
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Finally:
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<md>
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<mrow> \Cov(X, Y) = \E(XY) - \E(X)\E(Y) = 0.5 - (0.6)(0.7) = 0.08. </mrow>
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</md>
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</p>
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</statement>
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</example>
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<exercises xml:id="exercises-Covariance">
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<exercise>
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<statement>
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@@ -35,6 +35,7 @@
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<!-- other notation -->
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\Cov}{Cov}
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</macros>
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