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Finding joint pdf of independent variables.

  1. Nov 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose X1 and X2 are two independent gamma random variables, and X1~Gamma(a1, 1) and X2~(a2, 1).

    a) Find the joint pdf of Y1 = X1 + X2, and Y2 = X1/(X1 + X2).

    b) Show that Y1 and Y2 are independent.

    c) Find the marginal distributions of Y1 and Y2.

    3. The attempt at a solution

    a) 1/[Gamma(a1)Gamma(a2)] * (y1y2)a1 - 1 * (y1 - y1y2)a2 - 1 * e-y1 * y1

    b) Show joint pdf = product of individual pdfs of y1 and y2. To do this you have to find the marginal distributions so...

    c) To find the marginal distributions, integrate away the extra variable. So, for the pdf of y1 the integral you have to evaluate is...

    (e-y1y1/(Gamma(a1)Gamma(a2)) * integral of (y1y2)a1 - 1 * (y1 - y1y2)a2 - 1 dy2 as y2 goes from 0 to 1...

    But this integral is pretty hard to do by hand (unless I'm missing a trick). So is there a better way to do this problem?
     
  2. jcsd
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