Conditional expectation and total variance

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Problem

Let $Z \sim \operatorname{Bernoulli}(1/2)$ . Conditional on $Z=0$ , $X$ has mean $0$ and variance $1$ ; conditional on $Z=1$ , $X$ has mean $2$ and variance $3$ . Compute $\mathbb{E}[X]$ and $\operatorname{Var}(X)$ .

Reveal answer or reference solution

We have $\mathbb{E}[X]=1$ . Also,

\mathbb{E}[\operatorname{Var}(X\mid Z)] = \frac{1+3}{2} = 2

and

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

because $\mathbb{E}[X\mid Z]$ equals $0$ or $2$ with equal probability. Therefore, $\operatorname{Var}(X)=3$ .

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Probability