Product of two random variables

Hi guys,

Need a hint with the following problem:

Let $\displaystyle X$ and $\displaystyle Y$ be independent random variables, with probability distribution functions $\displaystyle f_{X}(x)$ and $\displaystyle f_{Y}(y)$. Let $\displaystyle Z= XY$ What is the probability distribution function for $\displaystyle Z$?

$\displaystyle F_{Z}(x) =P(Z \leq z) = P(XY \leq z) = P (Y \leq \frac{z}{x}) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{XY}(x, y)\,dy\,dx$

Since $\displaystyle X$ and $\displaystyle Y$ are independent, this gives:

$\displaystyle F_{Z}(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{X}(x)f_{Y}(y)\,dy\,dx$

Thus, we can obtain $\displaystyle f_{Z}(z)$ by differentiating both sides w/ respect to $\displaystyle z$ and using the fundamental theorem of calculus, which gives

$\displaystyle f_{Z}(z) = \int_{-\infty}^{\infty}\frac{1}{x}f_{X}(x) g_{Y}(\frac{x}{z})\,dx$

Thanks any help!

James

Re: Product of two random variables

Quote:

Originally Posted by

**james121515** Hi guys,

Need a hint with the following problem:

Let $\displaystyle X$ and $\displaystyle Y$ be independent random variables, with probability distribution functions $\displaystyle f_{X}(x)$ and $\displaystyle f_{Y}(y)$. Let $\displaystyle Z= XY$ What is the probability distribution function for $\displaystyle Z$?

$\displaystyle F_{Z}(x) =P(Z \leq z) = P(XY \leq z) = P (Y \leq \frac{z}{x}) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{XY}(x, y)\,dy\,dx$

Since $\displaystyle X$ and $\displaystyle Y$ are independent, this gives:

$\displaystyle F_{Z}(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{X}(x)f_{Y}(y)\,dy\,dx$

Thus, we can obtain $\displaystyle f_{Z}(z)$ by differentiating both sides w/ respect to $\displaystyle z$ and using the fundamental theorem of calculus, which gives

$\displaystyle f_{Z}(z) = \int_{-\infty}^{\infty}\frac{1}{x}f_{X}(x) g_{Y}(\frac{x}{z})\,dx$

Thanks any help!

James

http://www.math.wm.edu/~leemis/2003csada.pdf

Product distribution - Wikipedia, the free encyclopedia

Re: Product of two random variables

Hey,

Interesting, you have the same question as I have!

http://www.mathhelpforum.com/math-he...es-192173.html

I'm specifically looking for the case where X is exponentially distributed, and Y follows a normal distribution.

A closed-form solution seems impossible, hence I tried to estimate the pdf of Z.

Via Monte Carlo simulation, it appears that Z=XY, follows again an exponential distribution for z>0.