I have two RVs; their PDF are as the followings:
\begin{split} f_{X}(x) = \frac 1 {a} e^{-\frac x {a}}\end{split} and \begin{split} f_{Y}(y) = \frac {y^{L-1}} {b^{L} \Gamma (L)} e^{-\frac y {b}}\end{split}
where; a,b [Element] Reals && a,b > 0; L [Element] Integers && L >= 1;
Under the above conditions I want to calculated \begin{split} f_{Z}(z) \end{split} where \begin{split} Z = X/Y \end{split}
The PDF can be calculated as the follwing: $$f_Z(z) = \int_0^{\infty} y f_X(yz) f_Y(y) dy$$
And the CDF can be calculated as the follwing: $$F_Z(z) = \int_0^{\infty} \int_0^{yz} f_X(x) f_Y(y) dx dy$$ or $$F_Z(z) = \int_0^{\infty} F_X(yz) f_Y(y) dy$$
$$F_X(yz) = 1- e^{-\frac {yz} {a}} $$
Now to get the PDF of Z; the result for the above integration is: $$ f_Z(z) = \frac {a^L*b*L} {(a+b*z)^{L+1}} $$
And the CDF of Z is:
$$ F_Z(z) = 1-\left(\frac{a}{a+b z}\right)^L $$
How to check if these results are correct?
Simulate them. Write a quick computer program (Mathematica or Matlab (with Statistics toolbox) or octave) to simulate all three random variables, sample each 1000 times, and see if the associated empirical distributions are similar.