Expectation¶

Definition¶

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

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.

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.

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)\mathrm{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¶

• Random Variable
• Probability Mass Function
• Probability Density Function

Related Concepts¶

• Infinite Series
• Distribution
• Discrete Random Variable
• Survivor Function
• Indicator Random Variable
• Poisson Distribution
• Sample Mean

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