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Math Help - Probability Independence

  1. #1
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    Probability Independence



    A sequence of
    N independent Bernoulli trials is performed, where N in a non-
    negative integer-valued random variable, and the probability of success on any one trial is
    p. Let S be the total number of success and F be the total number of failures (S + F = N). Show that the joint probability generating function of S and F is given by GS,F (s,t) = GN(ps + (1 -p)t), where GN is the probability generating function of N.
    Hint: use conditional expectation, E( .) = E(E( . |N)).

    Also, show that if N has a Poisson distribution, then S and F are independent.


    What is E( . )? I dont recall ever seeing that. I have no chance in hell of doing this question without help

    Please Help!!

    Last edited by CaptainBlack; May 22nd 2008 at 06:57 AM.
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  2. #2
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    Quote Originally Posted by woody198403 View Post


    A sequence of
    N independent Bernoulli trials is performed, where N in a non-
    negative integer-valued random variable, and the probability of success on any one trial is p. Let S be the total number of success and F be the total number of failures (S + F = N). Show that the joint probability generating function of S and F is given by GS,F (s,t) = GN(ps + (1 -p)t), where GN is the probability generating function of N.
    Hint: use conditional expectation, E( .) = E(E( . |N)).

    Also, show that if N has a Poisson distribution, then S and F are independent.


    What in gods name is E( . )? I dont recall ever seeing that. I have no chance in hell of doing this question without help

    Please Help!!

    This might get you started (but I might be barking up the wrong tree):

    1. S ~ Binomial(N, p) and F ~ Bin(N, 1 - p).

    2. m_{S, F} (t_1, \, t_2) = E\left( e^{t_1 S + t_2 F}\right).

    Preliminary reading: http://www.stat.purdue.edu/~jingwu/S.../Lecture32.pdf



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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by woody198403 View Post

    What is E( . )? I dont recall ever seeing that. I have no chance in hell of doing this question without help


    E(.) is the expectation operator, it gives the average of a function of a RV.

    RonL

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