Tuesday, Feb 3

Section Tuesday, Feb 3

This is an outline of the topics we covered in class. These notes are not a substitute for your own note-taking. I highly recommend that you take your own notes during class. If you ever miss a class for any reason, reach out to another student in class to get a copy of their notes.

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Subsection Variance

Question: How spread out are \(X\) values? One answer we might try is to measure the average distance from the average value:

\begin{gather*} \E\left[|X - \E(X)|\right] \end{gather*}

To simplify notation, we’ll write \(\mu = \E(X)\text{.}\) Also, it’s often usefull to square a term rather than take absolute value when we want to ensure a positive output, so we’ll define...

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Definition 90.

Let \(X\) be a random variable with \(\E(X) = \mu\text{.}\) The variance of \(X\) is:

\begin{gather*} \sigma^2 = \Var(x) = \E\left[ (X - \mu)^2\right] \end{gather*}

\(\sigma = \sqrt{\Var(X)}\) is called the standard deviation.

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We can write an alternative formula here:

\begin{align*} \Var(X) \amp = \E\left[ (X - \mu)^2 \right] \\ \amp = \E\left[ X^2 - 2\mu X + \mu^2 \right] \\ \amp = \E(X^2) - 2\mu \E(X) + \E(\mu^2) \\ \amp = \E(X^2) - 2\mu^2 + \mu^2 \\ \amp = \E(X^2) - \mu^2. \end{align*}

This formula is generally more useful for performing computations.

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Example 91.

Let \(X\) indicate event \(A\) with \(\Pr(A) = p\text{.}\)

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\(k\)	\(\Pr(X = k)\)
\(0\)	\(1 - p\)
\(1\)	\(p\)

Table 92. Distribution for \(X\)

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\(k\)	\(\Pr(X^2 = k)\)
\(0^2\)	\(1 - p\)
\(1^2\)	\(p\)

Table 93. Distribution for \(X^2\)

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Then \(\E(X) = p\) and \(\E(X^2) = p\text{,}\) so:

\begin{gather*} \Var(X) = \E(X^2) - \left(\E(X)\right)^2 = p - p^2 = p(1 - p). \end{gather*}

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Example 94.

Let \(R\) be the roll of a fair D6.

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\(k\)	\(\Pr(R = k)\)
\(1\)	\(1/6\)
\(2\)	\(1/6\)
\(3\)	\(1/6\)
\(4\)	\(1/6\)
\(5\)	\(1/6\)
\(6\)	\(1/6\)

Table 95. Distribution for \(R\)

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\(k\)	\(\Pr(R^2 = k)\)
\(1^2\)	\(1/6\)
\(2^2\)	\(1/6\)
\(3^2\)	\(1/6\)
\(4^2\)	\(1/6\)
\(5^2\)	\(1/6\)
\(6^2\)	\(1/6\)

Table 96. Distribution for \(R^2\)

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Then \(\E(R) = \frac{7}{2}\) and \(\E(R^2) = \frac{91}{6}\text{,}\) so:

\begin{gather*} \Var(R) = \E(R^2) - \left(\E(R)\right)^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12}. \end{gather*}

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Example 97.

Let \(X \in [0, 1]\) with pdf \(f(x) = 2x\text{.}\)

\begin{align*} \E(X) \amp = \int_0^1 x \cdot 2x\ dx \\ \amp = \int_0^1 2x^2\ dx \\ \amp = \frac{2x^3}{3}\bigg|_0^1 \\ \amp = \frac{2}{3} - 0 \\ \amp = \frac{2}{3} \\ \E(X^2) \amp = \int_0^1 x^2\cdot 2x\ dx \\ \amp = \int_0^1 2x^3\ dx \\ \amp = \frac{2x^4}{4}\bigg|_0^1 \\ \amp = \frac{1}{2} - 0 \\ \amp = \frac{1}{2} \\ \Rightarrow \quad \Var(X) \amp = \E(X^2) - \left(\E(X)\right)^2 \\ \amp = \frac{1}{2} - \frac{4}{9} \\ \amp = \frac{1}{18}. \end{align*}

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WARNING!!! In general, variance is not linear! That is,

\begin{align*} \Var(X + Y) \amp \neq \Var(X) + \Var(Y) \\ \Var(kX) \amp \neq k \Var(X) \end{align*}

But there are still relevant properties we can state here:

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Theorem 98.

Let \(X\) be a random variable and \(k\in \R\text{.}\) Then:

\begin{align*} \Var(X + k) \amp = \Var(X) \amp \Var(kX) \amp = k^2\Var(X). \end{align*}

Moreover, if \(Y\) is another random variable and \(X, Y\) are independent, then:

\begin{gather*} \Var(X + Y) = \Var(X) + \Var(Y). \end{gather*}

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Example 99.

Let \(H_1, H_2, \dotsc, H_n\) be indicators with parameter \(p\text{.}\) Let \(S = H_1 + \dotsb + H_n\text{,}\) so \(S \sim \Bin(n, p)\text{.}\) We know \(\Var(H_i) = p(1-p)\text{,}\) and \(H_1, \dotsc, H_n\) are independent. So:

\begin{align*} \Var(S) \amp = \Var(H_1 + \dotsb + H_n) \\ \amp = \Var(H_1) + \dotsb + \Var(H_n) \\ \amp = p(1-p) + \dotsb + p(1-p) \\ \amp = np(1-p) \end{align*}

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