Here's a problem I am stuck on:
Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then:
- What is the CDF and PDF of $X+Y$ ?
- Does the PDF of $XY$ exist?
- What is the CDF of $XY$?
I tried finding the CDF of $X+Y$ by conditioning on $X$ as per this answer, but could not get any further. Can anyone show me what to do, or how to do this?