Find the median of a function of a normal random variable.

Question

If $X\sim N(\mu,\sigma^2)$ and $Y=e^X$, then what is the median of $Y$?

I am pretty sure that $Y$ is also distributed normal. To try to prove it, I attempted both the method of moment generating functions and the method of cdfs. I just can't get it. Thanks for your help. Once I show that, getting the median is the same as getting the mean.

Answer 1

Hints:

• What is a median of $X$ normal $N(\mu,\sigma^2)$?
• If $m$ is a median of the random variable $U$, and $V=A(U)$ where the function $A:\mathbb R\to\mathbb R$ is nondecreasing, what would be a median of $V$?

Answer 2

Hint: let $\Phi$ be the CDF of $Z \sim \mathcal{N}\left(0, 1\right)$. Notice that $$\mathbb{P}\left(Y \leq y\right) = \mathbb{P}\left(e^{X} \leq y\right) = \mathbb{P}\left(X \leq \ln(y)\right) = \mathbb{P}\left(Z \leq \dfrac{\ln(y)-\mu}{\sigma}\right)= \Phi\left[\dfrac{\ln(y)-\mu}{\sigma}\right]\text{.}$$