# Separating the product of two probability distributions

1. May 23, 2012

### pboggler

In general, how does one separate the product of two probability distributions with one of them known? Basically, I have the distribution of rcosθ, I know that P(cosθ) = 2/(πsinθ), and I want to find P(r). Wolfram Alpha makes me think that a delta function is involved based on what they say about uniform product distributions and normal product distributions, but I wouldn't know how to solve it in this case.

Here are the URLs to Wolfram Alpha's things. I can't include links until I make 10 posts apparently.
mathworld.wolfram.com/UniformProductDistribution.html
mathworld.wolfram.com/NormalProductDistribution.html

Thanks for any help or suggestions!

2. May 23, 2012

### mathman

Assuming r and θ are independent, you can set it up as follows (admittedly sketchy):

Let w = rcosθ, then P(r < R|cosθ) = P(w<Rcosθ|cosθ). Integrate over the distribution of θ.

3. May 24, 2012

### pboggler

What should I be integrating? And what will it tell me? Assuming the distributions are over [0,1], are you saying something like $P(w) = \int_0^1 P(w<Rcosθ|cosθ) d(cosθ)$?

4. May 24, 2012

### mathman

After integration you will end up with P(r<R).