Conditional Variance Given a Random Variable¶

Definition¶

For $\mathbb{E}[Y^2]<\infty$ $\infty$ , conditional variance is

\operatorname{Var}(Y\mid X) =\mathbb{E}[(Y-\mathbb{E}[Y\mid X])^2\mid X] =\mathbb{E}[Y^2\mid X]-(\mathbb{E}[Y\mid X])^2.

These identities hold almost surely.

Apply the definition of Variance and Standard Deviation within the law conditional on $X$ : subtract the Conditional Expectation Given a Random Variable and average its squared residual under that same conditional law.

Interpretation¶

Conditional variance measures how much spread in $Y$ remains once the value of $X$ is known. Like the conditional expectation $\mathbb{E}[Y\mid X]$ $X]$ , it is itself a random variable: a function of $X$ , not a single number.

The conditional mean line E(Y given X = x) = x with a band of one standard deviation, the square root of x, around it; the band is narrow at x = 1 and wider at x = 4.

The diagram shows $Y\mid X=x\sim\operatorname{Pois}(x)$ $Pois (x)$ . The band around the conditional mean line $\mathbb{E}[Y\mid X=x]=x$ has half-width $\sqrt{\operatorname{Var}(Y\mid X=x)}=\sqrt{x}$ $x$ : $\pm1$ at $x=1$ and $\pm2$ at $x=4$ $4$ , so larger $x$ gives more spread. Evaluating at the random $X$ gives the random variable $\operatorname{Var}(Y\mid X)=X$ $X$ , as in the first Example.

$\operatorname{Var}(Y\mid X=x)$ is the spread within group $x$ (right); averaged over $X$ it gives $\mathbb{E}[\operatorname{Var}(Y\mid X)]$ $X)]$ .

Read the definition in two steps. First fix $X=x$ and compute an ordinary variance under the conditional distribution of $Y$ given $X=x$ $x$ : for discrete $X$ this is the conditional variance given the event $\{X=x\}$ $x}$ , a number $v(x)$ ; for continuous $X$ the conditional PDF takes over this role. Then substitute the random $X$ to get $\operatorname{Var}(Y\mid X)=v(X)$ $v (X)$ . The squared deviations are measured from $\mathbb{E}[Y\mid X]$ $X]$ , the centre of each group, not from the overall mean $\mathbb{E}Y$ . The second formula is the familiar "mean of the square minus square of the mean", applied inside the conditional distribution.

Averaging $v(X)$ over $X$ gives $\mathbb{E}[\operatorname{Var}(Y\mid X)]$ $X)]$ , the within-group part of Eve's law. A common slip is to expect $\operatorname{Var}(Y)=\mathbb{E}[\operatorname{Var}(Y\mid X)]$ $X)]$ : this misses the spread of the group centres themselves, $\operatorname{Var}(\mathbb{E}[Y\mid X])$ $X])$ .

Key Results¶

$\operatorname{Var}(Y\mid X)\ge0$ almost surely.
For real constants $a,b$ , $\operatorname{Var}(a+bY\mid X)=b^2\operatorname{Var}(Y\mid X)$ $X)$ .

Examples¶

If $Y\mid X\sim\operatorname{Pois}(X)$ with $\mathbb{E}[X^2]<\infty$ $\infty$ , then $\operatorname{Var}(Y\mid X)=X$ $X$ .
If $Y\mid X\sim\operatorname{Bin}(n,X)$ with $0\le X\le1$ $1$ , then $\operatorname{Var}(Y\mid X)=nX(1-X)$ $X)$ .

Prerequisites¶

Conditioning Trick for Variance

Sources¶

Nicky van Foreest · Probability Distributions · Week 6

Symbols: infinity, square root

Last updated: 22 October 2026 19:04