Finding CDF and PDF of random variable Z=X/Y

Question

$$F_Z(z) = \int_0^{\infty} \int_0^{yz} f_X(x) f_Y(y) dx dy$$ for Y [Ymin,Ymax] and X>0 $$F_X(yz)F_Y(y) \Big|_Y^{Ymax} - \int_Y^{Ymax} F_Y(y) dF_X(yz) $$

Can I write an equation like this? Do you think I'm track in the right path?

Answer 1

As discussed in the reference here . One can obtain the pdf $f_Z(z)$ for the variable $z = g(x_1,...x_n)$ using the formula:

$ f_{Z}(z) = \int \,d^{n}x \,\,\delta(z-g(x_1,x_2,...,x_n)) f_{X}(x_1, ...,x_n) $

In your case $x_1=X$, $x_2=Y$, and $z=g(x,y)=x/y$. Depending on the functions, these integrals may be tough to do. Getting the CDF can be done by integrating over $z$ up to a point.

Unfortunately, I can't give you a more authoritative source of the above formula that has a proof. I am actually looking for that myself here.

Intuitively, what the insertion of the delta function does is restrict the volume of integration into the correct domain of the variables $x_i$.