Condition first, then remove conditioning

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Problem

Let $Z\sim\operatorname{Bernoulli}(1/4)$ . Conditional on $Z=0$ , $X$ has mean $0$ and variance $1$ ; conditional on $Z=1$ , $X$ has mean $4$ and variance $1$ . Find $E[X\mid Z]$ , $E[X]$ , and $\operatorname{Var}(X)$ .

Reveal answer or reference solution

The conditional mean is $E[X\mid Z]=4Z$ . Therefore

E[X]=E[4Z]=1

Also $E[\operatorname{Var}(X\mid Z)]=1$ , while

\operatorname{Var}(E[X\mid Z])=\operatorname{Var}(4Z) =16\left(\frac14\right)\left(\frac34\right)=3

By total variance, $\operatorname{Var}(X)=1+3=4$ .

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Condition first, then remove conditioning

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