Variance of a coin-gated payoff

MediumRandom Variables Statistics~5m

Alice's winnings $X$ have mean $12$ and variance $2$ . Bob then flips a fair coin: on heads he takes all of Alice's winnings, on tails he gets nothing. Let $Y$ be the amount Bob receives. Compute $\operatorname{Var}(Y)$ .

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Write $Y = B\,X$ with $B\sim\text{Bernoulli}(1/2)$ independent of $X$ ; note $B^2=B$ .
Get $\mathbb{E}[X^2]=\operatorname{Var}+\text{mean}^2=146$ , then $\operatorname{Var}(Y)=\tfrac12\mathbb{E}[X^2]-(\tfrac12\cdot12)^2$ .

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Variance of a coin-gated payoff

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