I'm studying for an exam at the moment, and these types of questions have just got me stumped to the point where I need a step-by-step walkthrough...
More specifically I've got two questions I just can't get past:
Given two random variables $X$ and $Y$ with
$ f_X(x)= \left\{ \begin{array}{l l} xe^{-x} & \quad \text{if $x$ > 0},\\ 0 & \quad \text{else}.\\ \end{array} \right. $
$ f_Y(y)= \left\{ \begin{array}{l l} e^{-y} & \quad \text{if $y$ > 0},\\ 0 & \quad \text{else}.\\ \end{array} \right. $
as respective densities, show that $Z = Y/X$ has the following distribution function
$ F_Z(z)= \left\{ \begin{array}{l l} 1-\frac1{(1+z)^2} & \quad \text{if $y$ > 0},\\ 0 & \quad \text{else}.\\ \end{array} \right. $
Also have to find the density function, but to my knowledge this is just deriving with respect to $z$ and is $\frac2{(z + 1)^3}.$
A very similar question asks to show that:
If $X, Y$ are random variables with given densities
$ f_X(x)=\frac12x^2e^{-x} \ \ if \ x >0, $
$ f_Y(y)=e^{-y} \ \ if \ y > 0, $
then $Z = X + Y$ has probability density function
$ f_Z(z)= \frac{z^3}6e^{-z}. $
I'm guessing the first step is to find $Z$'s distribution function, but this is the part that stumps me in the first question also. Please help.