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Math Help - gamma distribution

  1. #1
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    gamma distribution

    For Z distributed on R^3 with Z1, Z2, Z3 IID Z ~ exp(1)

    I have X = (Z2 + Z3)/(Z1 + Z2 + Z3)

    How can i determine the distribution of X and obtain its mean and variance?
    I was thinking perhaps I need to do something in terms of gamma because standard exponential is z ~ exp(1) iff z ~ G(1) and chi-square possibly???

    plz someone can help me??
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  2. #2
    Moo
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    Hello,

    As you said, recall that exp(1) ~ Gamma(1)
    Since X+Y ~ Gamma(a+b) if X ~ Gamma(a) and Y ~ Gamma(b), it follows that Z2+Z3 ~ Gamma(2)

    Now there's this thing that you should know : if X and Y are independent, and X ~ Gamma(a) and Y ~ Gamma(b), then X/(X+Y) ~ Beta(a,b)

    Hence, (Z2+Z3)/(Z1+[Z2+Z3]) follows a Beta distribution with parameters (2,1), because Z2+Z3 follows a Gamma(2) and Z1 follows a Gamma(1)
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  3. #3
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    thanks!!!
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