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Math Help - Expected value of random variables

  1. #1
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    Expected value of random variables

    Let N,X1, X2... be random variables where N has a poisson distribution with mean 3 while X1, X2...each has a Poisson distribution with mean 7.
    a) Determine E[N\sum^{N}_{i=1} X_i]
    b) Determine the variance of [\sum^{N}_{i=1} X_i]

    I have no idea where to start. Please give me a hand. Thanks
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by EitanG View Post
    Let N,X1, X2... be random variables where N has a poisson distribution with mean 3 while X1, X2...each has a Poisson distribution with mean 7.
    a) Determine E[\sum^{N}_{i=1} X_i]
    b) Determine the variance of [\sum^{N}_{i=1} X_i]

    I have no idea where to start. Please give me a hand. Thanks
    If you know no formula for this, the best way is to partition the set of events :
    (assuming that N and the sequence X_i are independent, and I guess there's an extra N in what you wrote... anyway, it would be the same reasoning)

    \mathbb{E}\left(\sum_{i=1}^N X_i\right)=\sum_{k=0}^\infty k \mathbb{P}\left(\sum_{i=1}^N X_i=k\right)

    And \mathbb{P}\left(\sum_{i=1}^N X_i=k\right)=\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^N X_i=k ~,~ N=n\right)

    =\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^n X_i=k ~,~ N=n\right)=\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^1 X_i=k\right)\mathbb{P}(N=n)

    The sum of Poisson variables is a Poisson variable (whose parameter is the sum of the parameters)
    And you have to see what happens if N is 0. This should be stated in your problem, or conventionally, the sum would be 0.


    It's a bit messy, but you can get the information you need from here...
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  3. #3
    MHF Contributor matheagle's Avatar
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    It looks like you have an extra N in the expectation.
    This is Wald's equation, see...
    Wald's equation - Wikipedia, the free encyclopedia
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  4. #4
    MHF Contributor matheagle's Avatar
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    typo in your bound

    Quote Originally Posted by Moo View Post
    Hello,

    If you know no formula for this, the best way is to partition the set of events :
    (assuming that N and the sequence X_i are independent, and I guess there's an extra N in what you wrote... anyway, it would be the same reasoning)

    \mathbb{E}\left(\sum_{i=1}^N X_i\right)=\sum_{k=0}^\infty k \mathbb{P}\left(\sum_{i=1}^N X_i=k\right)

    And P(\sum_{i=1}^N X_i=k)=\sum_{n=1}^\infty P(\sum_{i=1}^N X_i=k ~,~ N=n)

    =\sum_{n=1}^\infty P(\sum_{i=1}^n X_i=k ~,~ N=n)=\sum_{n=1}^\infty P(\sum_{i=1}^{n}X_i=k)P(N=n)

    The sum of Poisson variables is a Poisson variable (whose parameter is the sum of the parameters)
    And you have to see what happens if N is 0. This should be stated in your problem, or conventionally, the sum would be 0.


    It's a bit messy, but you can get the information you need from here...
    Follow Math Help Forum on Facebook and Google+

  5. #5
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    No guys that N is supposed to be in the first line. The quote ignored it for some reason
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