Conditional Variance Given an Event¶

Definition¶

Let $X$ be a random variable and let $B$ be an event with $\mathbb{P}(B)>0$$0$. The conditional variance of $X$ given $B$ is

provided the conditional expectations exist.

Interpretation¶

Conditional variance measures how much uncertainty about $X$ remains once the event $B$ is known to have occurred. It is the variance of $X$ computed with the conditional probabilities given $B$, around the conditional mean $\mathbb{E}[X\mid B]$$B]$.

In the figure $X$ takes the values $1,2,3,4$ with probabilities $0.1,0.2,0.3,0.4$, and $B=\{X\ge3\}$ leaves the values $3$ and $4$ with conditional probabilities $3/7$ and $4/7$ and conditional mean $25/7$. The arrows show the deviations $3-\tfrac{25}{7}=-\tfrac{4}{7}$ and $4-\tfrac{25}{7}=\tfrac{3}{7}$$\frac{3}{7}$; squaring them and weighting with the conditional probabilities gives

This is much smaller than $\operatorname{Var}(X)=1$$1$, because only two neighbouring values remain.

Conditioning on $B$ gives a new probability model, so every rule for variance holds after conditioning. In particular $\operatorname{Var}(X\mid B)=\mathbb{E}[X^2\mid B]-(\mathbb{E}[X\mid B])^2$$B]{)}^2$; here $\mathbb{E}[X^2\mid B]=9\cdot\tfrac{3}{7}+16\cdot\tfrac{4}{7}=13$ and $13-\tfrac{625}{49}=\tfrac{12}{49}$$\frac{12}{49}$. The centre must be the conditional mean: centring at $\mathbb{E}[X]=3$ instead would give $\tfrac{3}{7}\cdot0+\tfrac{4}{7}\cdot1=\tfrac{4}{7}$$\frac{4}{7}$, which is not the conditional variance.

Conditioning does not always reduce spread. Given $\{X=1\}\cup\{X=4\}$$4\}$, the two remaining values lie far apart, and the conditional variance is $1.44$, larger than $\operatorname{Var}(X)=1$$1$. Averaged over the events of a partition, however, conditional variances never exceed $\operatorname{Var}(X)$; this returns later in the law of total variance.

Key Results¶

• Conditional variance measures spread after conditioning on information.
• The law of total variance decomposes total variability into within-condition and between-condition parts.
• Conditional variance is zero when the condition determines the random variable exactly.

Examples¶

• Delivery time may have smaller variance after conditioning on the shipping method.
• Exam score variability may differ between students who studied and students who did not.
• In a mixture distribution, each component can have its own conditional variance.

Prerequisites¶

• Conditional Expectation
• Variance and Standard Deviation

Related Concepts¶

• Law of Total Probability
• Mixture Distribution

Sources¶

• Probability Theory for EOR · Week 4

Keywords: variance given information, conditional Var, Conditional Variance Symbols: union

Last updated: 20 October 2026 03:02