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Math Help - Probability Generating Functions Question

  1. #1
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    Red face Probability Generating Functions Question

    I'm just having trouble with this questions it's more proof then anything:

    Using probability generating functions verify the following results for independent random variables, and hence derive the expectation and variance of the sum.

    (a) If X1,....,Xk are Bernoulli random variables with common parameter p, then X1 +...+Xk has a binomial distribution with parameters (k,p)

    (b) If X1,...,Xk are geometric random variables with common parameter p, then X1+....Xk has a negative binomal distribution with parameters (k,p)

    (c) If X1,...,Xk are Poisson random variables with common parameters lambda1,....,lambdak, then X1+...+Xk also has a Poisson distribution with parameter lambda1+...lambdak.

    Don't really understand the question so any help would be appreciated.
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  2. #2
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    Quote Originally Posted by Hysterikz View Post
    I'm just having trouble with this questions it's more proof then anything:

    Using probability generating functions verify the following results for independent random variables, and hence derive the expectation and variance of the sum.

    (a) If X1,....,Xk are Bernoulli random variables with common parameter p, then X1 +...+Xk has a binomial distribution with parameters (k,p)

    (b) If X1,...,Xk are geometric random variables with common parameter p, then X1+....Xk has a negative binomal distribution with parameters (k,p)

    (c) If X1,...,Xk are Poisson random variables with common parameters lambda1,....,lambdak, then X1+...+Xk also has a Poisson distribution with parameter lambda1+...lambdak.

    Don't really understand the question so any help would be appreciated.
    Are the random variables independent?

    The probability generating function (PGF) of a sum of independent random variables is equal to the product of the PGF's of each random variable.

    So calculate the PGF of each sum and identify the resulting expression as the PGF of the given distribution.

    You should look up whatever PGF's you need to use. See Probability-generating function - Wikipedia, the free encyclopedia
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  3. #3
    MHF Contributor matheagle's Avatar
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    They have to be independent.
    You cannot determine the joint distribution from the marginals unless you know more about their joint structure.
    The study of their relationship is called a copula.
    http://en.wikipedia.org/wiki/Copula_(statistics)
    Abe Sklar used to be a colleague of mine and I always wondered
    why they didn't name one of their copulas...
    Francis Ford Copula.
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