Mathematics and Statistics · Ch 16 — Probability Distributions

Expected Value (Mean) and Variance

A distribution is summarised by two numbers: the expected value (mean), which locates its centre, and the variance, which measures its spread.

Note

Expected value (mean) $E(X)$

For a discrete variable with p.m.f. $p_i = P(X=x_i)$ ,

$E(X) = \mu = \sum_{i} x_i\, p_i .$

For a continuous variable with density $f(x)$ ,

$E(X) = \mu = \int_{-\infty}^{\infty} x\, f(x)\,dx .$

The expected value is the long-run average value of $X$ over very many repetitions of the experiment — a probability-weighted average of the values it can take.

Note

Variance and standard deviation

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

where $E(X^2) = \sum_i x_i^2\, p_i$ (discrete) or $\int x^2 f(x)\,dx$ (continuous). The standard deviation is $\sigma = \sqrt{\operatorname{Var}(X)}$ .

The identity $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$ is almost always the easier route in numerical work: compute $E(X)$ and $E(X^2)$ from the table, then subtract the square of the mean. Variance is never negative, and $\sigma$ carries the same units as $X$ , which is why it is the preferred measure of spread.

Tip

Useful properties (for constants $a, b$ )

$E(aX + b) = a\,E(X) + b$ and $\operatorname{Var}(aX+b) = a^2\,\operatorname{Var}(X)$ . Adding a constant shifts the mean but leaves the spread unchanged; scaling by $a$ scales the variance by $a^2$ . …

Definition 8Expected value (mean) $E(X)$

$E(X)=\sum x_i p_i$ (discrete) or $\int x f(x)\,dx$ (continuous): the probability-weighted average / long-ru …

Definition 9Variance $\operatorname{Var}(X)$

$\operatorname{Var}(X)=E(X^2)-[E(X)]^2$ , a non-negative measure of spread; its square root $\sigma$ is the …

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