Conditional Expectation Given an Event¶

Definition¶

Let $A$ be an event with $\mathbb{P}(A)>0$ and let $Y$ be integrable. Its conditional expectation given $A$ is the average under the conditional probability model:

\mathbb{E}[Y\mid A]=\frac{\mathbb{E}[Y\indicator_A]}{\mathbb{P}(A)}.

For discrete $Y$ , this equals $\displaystyle \sum_y y\,\mathbb{P}(Y=y\mid A)$ $A)$ . When the conditional law has a density $f_{Y\mid A}$ , it equals $\int_{\mathbb{R}}y f_{Y\mid A}(y)\,\mathrm{d}y$ . Conditioning on an event gives a number; conditioning on a random variable gives a random variable and is treated separately in Probability Distributions (conditional expectation given a random variable).

Interpretation¶

The conditional expectation $\mathbb{E}[Y\mid A]$ is the average value of $Y$ when only the outcomes in an event $A$ that is known to have occurred are considered. It is computed like an ordinary expectation, with the conditional probabilities given $A$ as weights.

Probability bars for X = 1 to 4 with masses 0.1 to 0.4, and the conditional bars given X at least 3, rescaled to 3/7 and 4/7; vertical lines mark the means 3 and 25/7.

In the figure $X$ takes the values $1,2,3,4$ with probabilities $0.1,0.2,0.3,0.4$ , so $\mathbb{E}[X]=3$ $3$ . Given $B=\{X\ge3\}$ $3}$ , with $\mathbb{P}(B)=0.7$ $0.7$ , the values $1$ and $2$ drop out, and the masses $0.3$ and $0.4$ are rescaled to $3/7$ and $4/7$ so that they again add up to $1$ . The conditional mean $\mathbb{E}[X\mid B]=3(3/7)+4(4/7)=25/7\approx3.57$ lies to the right of $3$ because only the large values remain.

In the formula $\mathbb{E}[Y\mid A]=\mathbb{E}[Y\indicator_A]/\mathbb{P}(A)$ $E [Y 𝟙 1_{A}] / P (A)$ , the indicator sets $Y$ to zero on the outcomes outside $A$ , so the numerator adds up $Y$ only over $A$ ; dividing by $\mathbb{P}(A)$ renormalizes those weights to total one. In the figure $\mathbb{E}[X\indicator_B]=3(0.3)+4(0.4)=2.5$ and $2.5/0.7=25/7$ $25/7$ . The condition $\mathbb{P}(A)>0$ is needed because the division is otherwise undefined. A common slip is to stop at $\mathbb{E}[Y\indicator_A]$ : here $2.5$ is smaller than every value $X$ takes on $B$ , so it cannot be the average over $B$ .

Conditioning on $A$ and on its complement and averaging the two results with weights $\mathbb{P}(A)$ and $\mathbb{P}(A^c)$ gives back $\mathbb{E}[Y]$ , the expectation version of the law of total probability. In the figure $\mathbb{E}[X\mid X\le2]=0.5/0.3=5/3$ $5/3$ , and $0.7\cdot\tfrac{25}{7}+0.3\cdot\tfrac{5}{3}=2.5+0.5=3$ $3$ .

Key Results¶

$\mathbb{E}[Y\mid A]=\mathbb{E}[Y\indicator_A]/\mathbb{P}(A)$ $E [Y 𝟙 1_{A}] / P (A)$ .
For constants $a,b$ and integrable $Y,Z$ , $\mathbb{E}[aY+bZ\mid A]=a\mathbb{E}[Y\mid A]+b\mathbb{E}[Z\mid A]$ $A]$ .

Examples¶

For a fair die conditioned to be even, the mean is $(2+4+6)/3=4$ $4$ .
For $Y\sim\operatorname{Unif}(0,1)$ conditioned on $Y>1/2$ $1/2$ , the mean is $\int_{1/2}^12y\,\mathrm{d}y=3/4$ $3/4$ .

Prerequisites¶

Sources¶

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

Aliases: conditional mean given an event Keywords: expected value given information, Conditional Expectation Symbols: integral, real numbers, sum

Last updated: 20 October 2026 03:02