1
$\begingroup$

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: $$ F(t) = \begin{cases} 0 & \mbox{for } t < 0\\ \tfrac{3}{4}t & \mbox{for } 0 ≤ t < 1\\ \tfrac{t}{4} + \tfrac{1}{2} & \mbox{for } 1 ≤ t < 2\\ 1 & \mbox{for } t ≥ 2\end{cases} $$

  • 3
    Find $P\{XY \leq t\}$ for a _fixed_ number $t$ between $0$ and $1$. You will need to use the law of total probability to account for the cases that $Y=1$ and $Y=2$. Repeat for the case when $t$ is between $1$ and $2$ (in which case life is a little simpler because you know for sure that $Y$ necessarily has value $2$).2012-11-09

1 Answers 1