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Math Help - Marginal Density (Please Help)

  1. #1
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    Marginal Density (Please Help)

    Find the marginal density of X given that (fx|y=y(x)) = e^(-y)*y^x/x!
    (i.e., that the conditional density of X, given that Y=y is Poisson(y))
    and that fy(y) = lamda^a / gamma(a) * y^(a-1)*e^(-lamda*y) where a(alpha) is a positive integer (i.e., the random variable Y has gamma(a,lamda) density). Hint: fx(x) is a well-known density.

    ???? ㅡㅡ;;
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  2. #2
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    Quote Originally Posted by ninano1205 View Post
    Find the marginal density of X given that (fx|y=y(x)) = e^(-y)*y^x/x!
    (i.e., that the conditional density of X, given that Y=y is Poisson(y))
    and that fy(y) = lamda^a / gamma(a) * y^(a-1)*e^(-lamda*y) where a(alpha) is a positive integer (i.e., the random variable Y has gamma(a,lamda) density). Hint: fx(x) is a well-known density.

    ???? ㅡㅡ;;
    The starting point would be

    f_X(x | y) = \frac{f(x, y)}{f_Y(y)} \Rightarrow f(x, y) = f_X(x | y) \, f_Y(y)

    where f(x, y) is the joint pdf of X and Y.

    Now substitute the given distributions for f_X(x | y) and f_Y(y).

    Now integrate f(x, y) with respect to y to get f_X(x).
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  3. #3
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    But the real problem is the integration.
    I cannot proceed from the ingral of f(x,y) with repect to y.
    Anythought?
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    Quote Originally Posted by dingdong View Post
    But the real problem is the integration.
    I cannot proceed from the ingral of f(x,y) with repect to y.
    Anythought?
    After substituting the various pdf's and simplifying, you have to deal with the following integral:

    \int_{0}^{+\infty} y^{\alpha + x - 1} e^{-(\lambda + 1) y} \, dy.

    I suggest making the substitution t = (\lambda + 1) y to get the integral representation of the gamma function.

    Note: \int_{0}^{+\infty} t^{\alpha + x - 1} e^{-t} \, dt = \Gamma (\alpha + x).
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