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Math Help - Confusion about combination of different distributions

  1. #1
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    Confusion about combination of different distributions

    Say that X has an exponential distribution with unknown lambda. Then I'm told that lambda has a gamma distribution. Now I'm asked to find the density of X. How do I approach this problem?
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  2. #2
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    It's called a mixture of distributions, and you solve it using integrals

    so we know f(X| \lambda) = \lambda e^{-\lambda x} and f(\lambda) = \frac{1}{\Gamma(\alpha) B^{\alpha}}x^{\alpha - 1} e^{-x/B} (assuming the parameters are alpha and beta).

    f(x) = \int^{\infty}_0 f(X| \lambda) f(\lambda) d \lambda
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  3. #3
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    How would I find E(X) and Var(X) without using the pmf just calculated?
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  4. #4
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    I don't think you can.

    Since E(x) = \int^{\infty}_0 xf(x)dx and same idea for Var(X).

    Also, since f(x) is continous, it's a PDF, not a PMF.
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