Calculating the cdf of $Z=XY$ given $\ln X, Y \sim \exp(1)$

Question

Let $X,Y$ be iid real random variables where $\ln X\sim \exp(1) $. How can I determine the cdf of $Z:=XY$? I would know how to do so for $X \sim \exp(1)$, but not sure how to deal with the $\ln.$

Some help would be much appreciated.

Answer 1

Hints:

• $\ln Z=\ln X+\ln Y$ where $\ln X$ and $\ln Y$ are iid and have standard exponential distribution.

• If you know the CDF of $\ln Z$ then you can also find the CDF of $Z=e^{\ln Z}$