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Math Help - limiting distribution of r.v s and chebyshev's inequality

  1. #1
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    limiting distribution of r.v s and chebyshev's inequality

    Hello everbody; i have a question. i really couldnt understand. i think it is easy for people who know theory. i have answer but help to walkthrough. i go on from the middle of question


    X ~ N(0,1)
    Y ~ chi_square(n)

    W_n = 1/n * X/Y

    answer says ; use chebyshev inequality

    E(Y/n) = 1
    Var(Y/n) =2/n < inf for all n
    n-> inf. lim V(Y/n) = 0

    so it says; W --> N(0,1) in distribution

    what i didnt understand is that where did the X go? is it ignored because it is standard normal distribution? or it seems it just reversed denominator.. i really didnt understand why.. thanks for your help.
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  2. #2
    MHF Contributor matheagle's Avatar
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    Don't you mean  {nX\over Y}

    since  {Y\over n}\to 1 by the SLLN.

    X doesn't go anywhere, it's free of n.
    But I think you wrote this incorrectly.
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