Covariance with a lognormal child

HardConditional Probability & Expectation Random Variables~9m

Let $X \sim \text{Exponential}(1)$ , and given $X = x$ let $Y$ be lognormal with $\log Y \sim \mathcal{N}(0,\ x)$ (mean $0$ , variance $x$ ). Compute $\operatorname{Cov}(X, Y)$ .

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The lognormal mean is $\mathbb{E}[Y\mid X]=e^{X/2}$ . Use the tower rule for $\mathbb{E}[Y]$ and $\mathbb{E}[XY]$ .
For $X\sim\text{Exp}(1)$ , $\mathbb{E}[e^{tX}]=1/(1-t)$ and $\mathbb{E}[Xe^{tX}]=1/(1-t)^2$ . Evaluate at $t=1/2$ .

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Covariance with a lognormal child

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