Variance under random parameters

HardRandom Variables Statistics~8m

Let $M$ and $S$ be independent $\text{Exponential}(1)$ random variables. Given $M = \mu$ and $S = \sigma$ , the pair $(X, Y)$ is bivariate normal with both means $\mu$ , both variances $\sigma^2$ , and correlation $\rho$ (a fixed constant with $|\rho| \le 1$ ). Compute $\operatorname{Var}(Y)$ .

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Law of total variance: $\operatorname{Var}(Y)=\mathbb{E}[\operatorname{Var}(Y\mid M,S)]+\operatorname{Var}(\mathbb{E}[Y\mid M,S])=\mathbb{E}[S^2]+\operatorname{Var}(M)$ .
For $\text{Exp}(1)$ : $\operatorname{Var}(M)=1$ , $\mathbb{E}[S^2]=2$ . ( $\rho$ doesn't affect $\operatorname{Var}(Y)$ .)

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Asked at: Citadel, Two Sigma

Variance under random parameters

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