S​uppose that $X$ is a continuous random variable with mean 1 and
variance 4. And let be $Y \sim N(0,1)$. Assume $X$ and $Y$ are independent.
Compute this:a) $\mathbb{E}(X \cdot \exp(Y))$
b) $\mathbb{V}(X \cdot \exp(Y))$
My solution
For a), I used the fact that $\exp(Y) \sim \operatorname{Lognormal}(0,1)$. Then we have:
\begin{align}
\mathbb{E}(X \cdot \exp(Y))&= \mathbb{E}(X)\cdot \mathbb{E}(\exp(Y))\\&=1 \cdot \exp\left (\frac{1}{2} \right )
\end{align}
For b), we have:
\begin{align}
\mathbb{V}(X \cdot \exp(Y))&= \mathbb{E} \left [ X^2 \cdot \exp(2Y) \right ]-\mathbb{E}^{2}(X \cdot \exp(Y)) \\&=\mathbb{E} \left [ X^2 \cdot \exp(2Y) \right ]-\exp(1)
\end{align}
But since this point I am not sure how can I continue. Do you have any idea? I would really appreciate your help. Thank you very much!
Best Answer
Use that $X^2$ and $\exp(2Y)$ are independent. Note that $2Y$ is also normal, so $\exp(2Y)$ is lognormal.