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Math Help - Linear combination of poisson random variables

  1. #1
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    Linear combination of poisson random variables

    Hi, can someone help me with this question:

    Let X~Poisson(1), Y~Poisson(2), define T = X + Y.
    a) Show that T~Poisson(3)
    b) Find the joint probability function of X and T
    c) Find the conditional distribution of X given T = n
    d) Compute Cov(X,T) and the correlation coefficient.

    I have no problems with a). But I got stuck at b) because I get that the jpf of X and T, f(X,T) = P ( X = x , T = t) = f(T) ...?! Which makes no sense. Any help would be appreciated!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Assuming independence....

    Use moment generating functions to show (a).

    P(X=a,T=b)=P(X=a, Y=b-a)=P(X=a)P(Y=b-a)

    = \left({e^{-1}1^a\over a!}\right) \left({e^{-2}2^{b-a}\over (b-a)!}\right)

    for (c) just use the definition of conditional probabilities.

    (d) Cov(X,X+Y)=Cov(X,X)+Cov(X,Y)=V(X)+0=1.
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  3. #3
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    ah yes, thank you very much! So the difference between a) and b) is that in b) we are not summing all the marginal probabilities of x and y, because we have a constraint on what X is, correct?
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