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Math Help - consitent estimator

  1. #1
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    consitent estimator

    let Xn , n= 1, 2, . . . . . be a sequence of r.v st

    p(Xn = a) = 1 - (1/n), a in R

    p(Xn = n^k) = 1/n k>1 , k fixed

    show Xn is consitent a consistant estimator for a.

    i.e show for all e>0 p(|Xn - a|>= e) tends to 0 as n tends to infinity

    any idears

    thanks for you help
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  2. #2
    MHF Contributor matheagle's Avatar
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    The probability of that event is 1/n, which clearly goes to zero...

    P(|X_n-a|>\epsilon)\le P(X_n\ne a)=1/n\to 0
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  3. #3
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    i kinda see i dont really understand why the first inequality is true though
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  4. #4
    MHF Contributor matheagle's Avatar
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    Look at the number line....

    (-\infty,a-\epsilon)\cup(a+\epsilon,\infty)

    is contained in the interval

    (-\infty,a)\cup(a,\infty)

    Hence it has smaller probability.
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  5. #5
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    ok i see thank you
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