Here's the question
Let X and Y be independent random variables, where X is uniformly distributed over (2,4) and Y is exponentially distributed with mean 3. Find the density of U=X/Y
Here's what I got: (3/2)e^(-u/3)
I tried doing the PDF of both of them and divide them subbing in U for Y and it seems that's not the way to do it. I asked my professor for help and he told me that "I was on the right track and to keep thinking about it like that." However, I am at a complete loss as to what exactly I'm suppose to do.
Can anyone help me out with this one?
Best Answer
Let $U=X/Y$ Then since $X$ and $Y$ are strictly positive.
$$\begin{align}F_U(u)~=~&\Pr(U\leq u) \\[1ex] ~=~& \mathsf P(Y\geq uX) \\[1ex] =~& \int_2^4 \mathsf P(Y\geq X/u\mid X=x)~f_X(x)\operatorname d x \\[1ex] =~& \tfrac 1 2\int_2^4 \mathsf P(Y\geq x/u)\operatorname d x\quad\mathbf 1_{u\in(0;\infty)} \\[1ex] =~& \tfrac 1 2\int_2^4 \mathsf e^{-x/3u}\operatorname d x\quad\mathbf 1_{u\in(0;\infty)} \\[3ex] f_U(u) ~=~& \dfrac{\operatorname d~F_U(u)}{\operatorname d~u\qquad}\end{align}$$