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.
Best Answer
There is a "lather rinse repeat" way of solving such problems. I'll show you some steps to do the first one, you can try the technique on the second one too.
\begin{eqnarray*} F_Z(z) &=& \Pr(Z \leq z) \\ &=& \Pr(Y/X \leq z) \\ &=& \int_0^\infty \Pr(Y \leq zx) f_X(x) dx \\ &=& \int_0^\infty (1-e^{-zx})xe^{-x} dx. \end{eqnarray*}
Run this integral through, and that should do it. Of course, some of these steps may require careful justifications: for instance going from $Y/X \leq z$ to $Y \leq zX$ requires care with the sign of $X$. I hope you can figure these details out.
Alternatively, for the second one, you may want to try going the moment generating function/characteristic function route.