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Math Help - Convergence in distibution of a product

  1. #1
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    Convergence in distibution of a product

    I'm having afew troubles with the following questions, any help would be greatly appreciated.

    Q1. Suppose X1,X2,...,Xn are iid random variables, each uniformly distributed on the interval [0,1]. Calculate the mean and variance of log(X1).
    - i think i've done this although i can't help but get a variance of -3 ? getting a mean of -1.

    suppose that 0=< a =< b, show that
    P((X1X2....Xn)^1/sqrt(n) in [a,b]) tends to a limit as n tends to infinity and find an expression for it.

    -I'm not sure what to do for this, maybe something to do with using the log, as the log of a product is the sum of the logs..

    Q2. Poisson sampling
    Suppose X1,X2,...,Xn are iid random variables, each with the poisson distribution of parameter lambda>0.
    Let X' =(X1+X2+...Xn)/n, find the support and pmf of X'.

    - Now i obviously know the pmf of X1,X2 etc.. but not quite sure what to do when you sum random variables. my idea was to add the pmf of all the Xi's, but use k=k/n instead. where pmf=P(X=k).

    any help? thanks alot
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  2. #2
    MHF Contributor matheagle's Avatar
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    1 Show your work.
    2 Use the fact that the sum of Poisson's is a Poisson, then divide by n.
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  3. #3
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    1. for the mean i did the integral between 0 and 1 of log x, as the pdf of X is 1. so that gives xlog x - x between 0 and 1, which a little ropeily gives mean = -1.
    Then for the variance i did it as the integral of (log x)^2 then - (-1)^2. which i'm not sure is the best way to do it.
    Then the next bit i'm not really sure how to go about, is the central limit theorem anything to do with it?

    2. So is that just the pmf of the normal poisson distribution, but with nlambda instead of lambda. and then just divide the whole pmf by n?
    I really should know this stuff to be honest, my basic knowledge has just sort of escaped me
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  4. #4
    MHF Contributor matheagle's Avatar
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    In the first I would obtain the denisty of Y= log X.
    That's pretty easy.
    I think -log X is a chi-square or exponential, something basic.
    From that you can recognize the mean and variance.
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  5. #5
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    Okay thanks, i'd tried doing it that way with the density of Y=log X but didn't really see the mean and variance. can anyone shed any light on any of the other things?
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  6. #6
    MHF Contributor matheagle's Avatar
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    IF you recognize that -log X is a KNOWN distribution you need not perform
    any more calculus. Those distribution's means and variances have already been calculated.
    AND again show me the density of log X or -log X
    -----------------------------------------------------
    If X\sim U(0,1), then Y=-\ln X\sim exp (1)

    So the mean of -Y is -1 and variance is 1.
    Last edited by matheagle; December 23rd 2009 at 11:06 PM.
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  7. #7
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    oh sorry, i meant i'd tried it before posting anything on here when i wasn't really looking for the mean and variance properly. cheers for that anyway though
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  8. #8
    MHF Contributor matheagle's Avatar
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    and that product of X1 through Xn....
    Take the log, the product becomes a sum.
    Each log Xi is the negative of an exp(1), so the sum is the negative of a gamma.
    AND that does NOT mean the density is negative.
    The support is x<0 though.
    Last edited by matheagle; December 23rd 2009 at 11:06 PM.
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