Variance of Discrete Random Variables

Defines the variance of a discrete random variable as the expected squared deviation from the mean, develops the shortcut formula Var(X) = E[X^2] - (E[X])^2, and establishes the linear transformation property Var(aX+b) = a^2 Var(X).

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Tutorial

Defining the Variance

The variance of a discrete random variable $X$ measures the average squared deviation of $X$ from its mean $\mu = E[X]$ . With $p(x) = P(X=x),$ it is defined as

\text{Var}(X) = E[(X-\mu)^2] = \sum\limits_x (x-\mu)^2 \, p(x).

A small variance means the values of $X$ cluster tightly around $\mu;$ a large variance means they spread widely. Because each term $(x-\mu)^2 p(x)$ is non-negative, $\text{Var}(X) \geq 0.$

To illustrate, suppose $X$ takes values $0,$ $2,$ and $4,$ each with probability $\dfrac{1}{3}.$ The mean is

\mu = 0 \cdot \tfrac{1}{3} + 2 \cdot \tfrac{1}{3} + 4 \cdot \tfrac{1}{3} = 2.

The variance is then

\begin{align*} \text{Var}(X) &= (0-2)^2 \cdot \tfrac{1}{3} + (2-2)^2 \cdot \tfrac{1}{3} + (4-2)^2 \cdot \tfrac{1}{3} \\[3pt] &= \tfrac{4}{3} + 0 + \tfrac{4}{3} \\[3pt] &= \tfrac{8}{3}. \end{align*}