Conditional Variance for Discrete Random Variables

Compute the conditional variance $\text{Var}(X \mid Y=y)$ of a discrete random variable $X$ given the event $Y=y$ , using both the definitional formula and the computational identity $\text{Var}(X \mid Y=y) = E[X^2 \mid Y=y] - (E[X \mid Y=y])^2$ . Apply these to conditional pmfs and to joint pmf tables.

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Tutorial

Introduction

Let $X$ and $Y$ be discrete random variables. The conditional variance of $X$ given the event $Y=y$ is the variance of $X$ computed using the conditional distribution $P(X=x \mid Y=y){:}$

\text{Var}(X \mid Y=y) = \sum_x \big(x - E[X \mid Y=y]\big)^2 \cdot P(X=x \mid Y=y).

It measures the spread of $X$ around its conditional mean $E[X \mid Y=y]$ , once we know that $Y=y$ .

To illustrate, let $X$ be the value shown on a fair six-sided die, and condition on the event that the value is even. The conditional pmf is

P(X=2 \mid \text{even}) = P(X=4 \mid \text{even}) = P(X=6 \mid \text{even}) = \dfrac{1}{3},

so the conditional mean is

E[X \mid \text{even}] = \dfrac{2+4+6}{3} = 4.

Applying the definition,

\text{Var}(X \mid \text{even}) = \dfrac{(2-4)^2 + (4-4)^2 + (6-4)^2}{3} = \dfrac{4+0+4}{3} = \dfrac{8}{3}.