# Density function of product of random variables

1. May 7, 2014

### khotsofalang

suppose you have two random variables X and Y which are independent,
we want to form a new random variable Z=XY, if f(x) and f(y) are density functions
of X and Y respectively what is the density function of Z?

I tried taking logs and applying convolution, but it did not really work

2. May 7, 2014

$$P(Z \le z) = E_Y \left[P\left( XY \le z \mid Y = y\right) \right]$$