Expectation¶

Definition¶

For an integrable real-valued random variable $X$ , its expectation is the probability-weighted mean. The formulas are

\begin{aligned} \mathbb{E}X&=\sum_x x\,p_X(x) \quad\text{for discrete }X,\\ \mathbb{E}X&=\int_{\mathbb{R}}x f_X(x)\,\mathrm{d}x\quad \text{when }X\text{ has a density}. \end{aligned}

Finite expectation requires $\mathbb{E}|X|<\infty$ $\infty$ : respectively $\displaystyle \sum_x|x|p_X(x)<\infty$ or $\int_{\mathbb{R}}|x|f_X(x)\,\mathrm{d}x<\infty$ $\infty$ . More generally, write $X=X^+-X^-$ with positive part $X^+=\max(X,0)$ and negative part $X^-=\max(-X,0)$ $max (- X, 0)$ ; the extended expectation is defined if at least one of $\mathbb{E}X^+$ and $\mathbb{E}X^-$ is finite. If both are infinite, $\mathbb{E}X$ is undefined.

Interpretation¶

Expectation describes the centre of a distribution by weighting each outcome by its probability. Sums and integrals express the same idea for discrete variables and variables with densities.

Probability bars at 1, 2, 3 and 4 with masses 0.1 to 0.4, balanced on a fulcrum at the mean 3.

For the illustrated masses, the mean is $\mathbb{E}[X]=1(0.1)+2(0.2)+3(0.3)+4(0.4)=3$ $3$ . This probability-weighted average is the balance point of the masses and the long-run average of i.i.d. draws.

Treat the probabilities as weights on a seesaw and move the fulcrum $c$ : it balances exactly at $\displaystyle c=\mathbb{E}X=\sum_x x\,p_X(x)$ $x \sum x p_{X} (x)$ .

For a discrete variable, the probability mass function supplies the weight attached to each possible value in the expectation sum. These are probability weights, not equal weights unless the distribution is uniform.

Key Results¶

For nonnegative $X$ , expectation is well-defined in $[0,\infty]$ , even when it is not finite.
For $X\ge0$ $0$ , $\mathbb{E}X=\int_0^\infty\mathbb{P}(X>t)\,\mathrm{d}t$ $t) d t$ , possibly $+\infty$ .

Examples¶

For a fair die, $\mathbb{E}X=(1+2+3+4+5+6)/6=3.5$ $3.5$ .
If $f_X(x)=2x$ for $0<x<1$ and zero elsewhere, then $\mathbb{E}X=\int_0^1 2x^2\,\mathrm{d}x=2/3$ $2/3$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 4
Nicky van Foreest · Probability Distributions · Refresher
Nicky van Foreest · Probability Distributions · Refresher

Aliases: expected value, expectation value, mathematical expectation, population mean Keywords: mean, first moment, average, Expectation of a Discrete Random Variable, Expectation of a Continuous Random Variable Symbols: infinity, integral, real numbers, sum

Last updated: 22 October 2026 19:04