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Math Help - E[ | X - Y | ] where X and Y are independent Poisson random variables

  1. #1
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    E[ | X - Y | ] where X and Y are independent Poisson random variables

    What is the expected value of the absolute difference of two independent Poisson variables?

    E[ |X - Y| ]

    I've split the double sum into the correct regions but not sure what to do with the partial sums remaining.

    I have:

    Sum_0^infinity p(x) Sum_0^infinity |X - Y| p(y)

    ...since p(x,y) = p(x)p(y)

    = Sum_0^infinity p(x) [Sum_0^x (X - Y) p(y) + Sum_x^infinity (Y - X) p(y)]

    Should get something like | E[X] - E[Y] | + some variance or covariance term, the latter of which will be 0 since X and Y are independent.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

    what are the means of these Poisson's?
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    Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

    Was going to write the answer in terms of E[X] and E[Y] but if not could use:

    mean of X : E[X] = \lambda^X dt
    mean of Y : E[Y] = \lambda^Y dt
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    MHF Contributor matheagle's Avatar
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    Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

    I dont follow
    X is a random variable, the means are random?
    And the dt is?
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    Re: E[ | X - Y | ] where X and Y are independent Poisson random variables

    No the means are not random, apologies for confusion caused.

    Ok, lets just say the means are:

    mean X = E[X] = m
    mean Y = E[Y] = n

    I think the problem can be done in general for any Random variables X and Y.
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