LOTUS¶

Theorem¶

Let $g$ be a Borel-measurable real function (preimages of Borel sets are Borel sets), so that $g(X)$ is a random variable. If $\mathbb{E}|g(X)|<\infty$ $\infty$ , the law of the unconscious statistician (LOTUS) gives

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

The formulas also hold as extended nonnegative expectations when $g\ge0$ $0$ . A finite signed result requires absolute integrability; no distribution of $g(X)$ has to be computed first.

Interpretation¶

To average a transformed variable, apply the transformation to each possible value and weight using the original distribution. This avoids deriving a new distribution merely to compute an expectation.

The parabola g(x) = x squared with points at x = -1, 0, 1 and 2, and the probabilities 0.1, 0.3, 0.4 and 0.2 of these values written below.

For the transformation $g(x)=x^2$ shown, $\mathbb{E}[g(X)]=1(0.1)+0(0.3)+1(0.4)+4(0.2)=1.3$ $1.3$ . The values of $g$ are weighted by the PMF of $X$ , without deriving the PMF of $g(X)$ .

A function of a random variable changes the value on each outcome from $X$ to $g(X)$ . LOTUS averages those transformed values using the original law of $X$ , avoiding a separate derivation of the distribution of $g(X)$ .

Key Results¶

Choosing $g(x)=x^k$ computes the $k$ th moment when it exists; in particular $g(x)=x^2$ gives the second moment.
Choosing $g(x)=\indicator_{\{x\in A\}}$ gives $\mathbb{P}(X\in A)$ by a sum or an integral.

Examples¶

For a fair die, $\mathbb{E}[X^2]=(1+4+9+16+25+36)/6=91/6$ $91/6$ .
For $X\sim\operatorname{Exp}(2)$ $Exp (2)$ , $\mathbb{E}[\mathrm{e}^{-X}]=\int_0^\infty2\mathrm{e}^{-3x}\,\mathrm{d}x=2/3$ $2/3$ .

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 4
Nicky van Foreest · Probability Distributions · Refresher
Nicky van Foreest · Probability Distributions · Week 2

Aliases: law of the unconscious statistician Keywords: expectation of a function of a random variable, LOTUS for a Discrete Random Variable, LOTUS for a Continuous Random Variable Symbols: element of, infinity, integral, real numbers, sum

Last updated: 22 October 2026 19:04