diff --git a/source/ch-Expected-Value.ptx b/source/ch-Expected-Value.ptx index ae04669..38170fe 100644 --- a/source/ch-Expected-Value.ptx +++ b/source/ch-Expected-Value.ptx @@ -538,9 +538,55 @@ Variance

- Text of section. + The expected value is one of several statistics sometimes referred to as a "measure of central tendency". + These are statistics that try to capture the "typical" value of the random variable. + Our next statistic tries to measure how spread out a distribution is.

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+ Let X be a random variable. + The variance of X is: + + \Var(X) = \E\left[ \left(X - \E(X)\right)^2 \right]. + +

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+ This formula tries to capture some measure of how spread out the distribution is. + One way we might hope to measure spread is to calculate how far away X is from its expected value on average. + Squaring the term X - \E(X) ensures that any difference between the value of the random variable and its expected value contributes positively. + ("Distance away from" should be positive.) So, the formula in is conceptually clear. + In practice, there's another way to structure the formula that makes for cleaner calculation: +

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+ Let X be a random variable. + Then: + + \Var(X) = \E\left(X^2\right) - \left(\E(X)\right)^2. + +

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+ + \Var(X) \amp = \E\left[ \left(X - \E(X)\right)^2 \right] + \amp = \E\left[ X^2 - 2 \E(X) X + \left(\E(X)\right)^2\right] + \amp = \E\left(X^2\right) - 2 \E(X) \E(X) + \left(\E(X)\right)^2 + \amp = \E\left(X^2\right) - \left(\E(X)\right)^2. + +

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