Expected value and variance of a linearly transformed CRV

Consider the continuous random variable \(X\) and let \(a, b\) be non-zero constants.

\[\text{Then}\hspace{0.5em} \mathrm{E}(aX + b) = a\mathrm{E}(X) + b \text{ and}\hspace{0.5em} \mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X).\]

Proof:

\(\begin{aligned} \mathrm{E}(aX + b) &= \int_{\text{all}\hspace{0.5em} x} (ax + b) \mathrm{f}(x) \, \mathrm{d}x \\ &= \int_{\text{all}\hspace{0.5em} x} ax\mathrm{f}(x) + b\mathrm{f}(x) \, \mathrm{d}x \\ &= a \mathrm{E}(X) + b \end{aligned}\)

\(\begin{aligned} \mathrm{Var}(aX + b) &= \mathrm{E}\left((aX + b)^2\right) - [\mathrm{E}(aX + b)]^2 \\ &= \mathrm{E}(a^2X^2 + 2abX + b^2) - (a\mathrm{E}(X) + b)^2 \\ &= a^2\mathrm{E}(X^2) + 2ab\mathrm{E}(X) + b^2 - (a^2[\mathrm{E}(X)]^2 + 2ab\mathrm{E}(X) + b^2) \\ &= a^2\mathrm{E}(X^2) - a^2[\mathrm{E}(X)]^2 \\ &= a^2 \mathrm{Var}(X) \end{aligned}\)

This result applies to both discrete and continuous random variables.

The affine rules also hold when either constant is zero. If \(a=0\), then \(aX+b\) is the constant \(b\), with mean \(b\) and variance zero. For any real \(a\), the standard deviation of \(aX+b\) is \(|a|\operatorname{SD}(X)\).

When converting an event involving \(Y=aX+b\) back to an event in \(X\), reverse the inequality if dividing by a negative \(a\). Transform the event as well as the mean and spread. These formulas concern one variable under a change of scale and origin.