If $X$ is uniform on $[0, 1]$, and $Y$ is a discrete random variable which is either $1$ or $2$ with probability $\frac12$ each, find the cumulative distribution function of the product $XY$.
I know what the answer to the question is, I simply want to know HOW to solve this. The answer is: $ F(t) = \begin{cases} 0 & \mbox{for } t < 0\\ \tfrac{3}{4}t & \mbox{for } 0 ≤ t < 1\\ \tfrac{t}{4} + \tfrac{1}{2} & \mbox{for } 1 ≤ t < 2\\ 1 & \mbox{for } t ≥ 2\end{cases} $