0
$\begingroup$

I need help to compute $E[e^Y|X]$. Here both $X$ and $Y$ are standard normal random variables with mean $0$ and variance $1$, and are not necessarily independent. The answer will presumably depend on the coefficient of correlation $\rho=E[XY]$. I thought about this for a while but couldn't come up with anything.. any hints?

Thanks!

  • 1
    The correlation of two standard normal distributions does not uniquely determine how they may depend on each other. – 2017-02-22

2 Answers 2

1

Note that you can't do anything unless you know that the pair $(X,Y)$ is a bivariate normal, which is not always true even if $X,Y$ are normal. In that case you can write $X=\rho Y+\sqrt{1-\rho^2}Z$ where $Z$ is a standard normal independent of $Y$, or $Y=\rho X+\sqrt{1-\rho^2}W$ where $W$ is independent of $X$, whatever is easier. In this case $\rho=\text{Cov}(X,Y)$. Then you just need to know $\mathbb{E}[e^{\theta Z}]$ for a standard normal $Z$, this is the MGF of $Z$ evaluated at $\theta$, and is equal to $e^{\frac{\theta^2}{2}}$.

1

Think about how you can write $Y$ in term of $X$ and $Z$ where $Z$ is also $N(0,1)$ and independent of $X$.