Probability - Finding the Cumulative Distribution Function 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:
$$
F(t) = \begin{cases}
0 & \mbox{for } t < 0\\
\tfrac{3}{4}t & \mbox{for } 0 ≤ t < 1\\       
\tfrac{t}{4} + \tfrac{1}{2} & \mbox{for } 1 ≤ t < 2\\
1 & \mbox{for } t ≥ 2\end{cases}
$$

As Dilip had outlined, you the law of total probability:
$$\begin{eqnarray}
    \mathbb{P}\left(X Y \leqslant t\right) &=& \mathbb{E}\left( \mathbb{P}\left(X Y \leqslant t | Y\right)\right) = \mathbb{E}\left( \mathbb{P}\left(X \leqslant \frac{t}{Y} | Y\right)\right)
 =\mathbb{E}\left( F_X\left(\frac{t}{Y}\right)\right) \\&=& F_X\left(t\right) \mathbb{P}\left(Y=1\right) + F_X\left(\frac{t}{2}\right) \mathbb{P}\left(Y=2\right) \\
  &=& F_X(t) + F_X\left(\frac{t}{2}\right)
\end{eqnarray}
$$
It now remains to combine piecewise expressions for $F_X(t) = \max(0,\min(1,t))$.
