Properties of Variance for Discrete Random Variables

Use the computational formula Var(X)=E[X^2]-(E[X])^2, the linear transformation rule Var(aX+b)=a^2 Var(X), and the additivity of variance for independent random variables Var(X±Y)=Var(X)+Var(Y).

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The Computational Formula for Variance

The variance of a discrete random variable $X$ is defined as $\operatorname{Var}(X) = E[(X-\mu)^2],$ where $\mu = E[X].$ Expanding the square and applying linearity of expectation gives an equivalent — and usually faster — computational formula:

\operatorname{Var}(X) = E[X^2] - (E[X])^2.

The derivation is short:

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

To illustrate, suppose $X$ has the PMF

\begin{array}{c|ccc} x & 0 & 1 & 2 \\ \hline P(X=x) & 0.5 & 0.3 & 0.2 \end{array}

Then

\begin{align*} E[X] &= 0(0.5) + 1(0.3) + 2(0.2) = 0.7, \\ E[X^2] &= 0^2(0.5) + 1^2(0.3) + 2^2(0.2) = 1.1, \\ \operatorname{Var}(X) &= 1.1 - 0.7^2 = 1.1 - 0.49 = 0.61. \end{align*}