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Math Help - marginal distribution

  1. #1
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    marginal distribution

    Let X \;and \; Y be random variables. Suppose that the conditional distribution of X \; given \; Y=y is binomial distribution with parameters n \;and \; y. Assume that Y is a continuous \; uniform\; (0,1) distribution. Find the marginal distribution of X.

    I think that:

    since, P(X|Y) = \dfrac{f(x,y)}{f_{Y}(y)}

    but f_{Y}(y) = 1

    so, f(x,y) = \text{Binomial}(n,y)

    is this correct? I am confused on finding the marginal pdf. Can anyone help?
    Last edited by chutiya; November 1st 2010 at 11:28 PM.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Use the beta density.
    The answer is a discrete uniform distribution over the set 0,1,2..., n, which makes sense.

    The joint distribution is f(x,y)={n\choose x}y^x(1-y)^{n-x} I(0\le y\le 1)

    Next integrate out the y to get the marginal of X and use the fact that

    \int_0^1 y^a(1-y)^bdy={\Gamma (a+1)\Gamma(b+1)\over \Gamma(a+b+2)}

    since x and n are integers this reduces to 1/(n+1) which is correct on the set for x=0,1...,n.

    Initially I tried the MGF where I conditioned on the Y, but the MGF of the uniform is worthless, so I went the direct route.
    Last edited by matheagle; November 4th 2010 at 12:37 AM.
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  3. #3
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    thiat was exactly how I did it! Thanks!!
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