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Separating the product of two probability distributions

  1. May 23, 2012 #1
    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. jcsd
  3. May 23, 2012 #2

    mathman

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    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.
     
  4. May 24, 2012 #3
    What should I be integrating? And what will it tell me? Assuming the distributions are over [0,1], are you saying something like [itex]P(w) = \int_0^1 P(w<Rcosθ|cosθ) d(cosθ)[/itex]?
     
  5. May 24, 2012 #4

    mathman

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