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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.

  • 0
    For the cumulative probability distribution function of a random variable called capital $X$, one should write $F_X$, with a subscript capital $X$. Its value at a particular number called lower-case $x$ would then be $F_X(x)$. For example, its value at $3$ is $F_X(3)$ (_not_ $F_3(3)$, which is tacitly suggested by writing $F_x(x)$, as if the subscript and the argument in parentheses were both the same thing). And similarly for density functions.2012-08-20
  • 2
    Does it say anywhere that $X$ and $Y$ are **independent** random variables? Without this (or a specification of the _joint_ density $f_{X,Y}(x,y)$), the problem cannot be solved.2012-08-20
  • 0
    A very similar question (with the independence assumption mentioned by @Dilip) is solved at length [here](http://math.stackexchange.com/a/30966/6179).2012-08-20
  • 0
    @did thanks for the comment. As I didnt solve explicitely I dont know but I believe in you.2012-08-20

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