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Jerry asked in 科學數學 · 8 years ago

連續隨機變量問題!

Suppose that X is a continuous random variable with density f and cdf F.

We say that X has tail-index a if

limx->∞ x^a.[1 - F(x)] = A;

for some positive, finite number A.

Prove that X has tail-index a if and only if

limx->∞ x^a+1.f(x) = aA:

1 Answer

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  • 8 years ago
    Favorite Answer

    1. a > 0

    If a <= 0 then lim(x->∞) x^a [1-F(x)] = 0, contradicts to the condition A > 0.

    2. By L' Hopital rule

    lim(x->∞) x^a [1-F(x)]

    =lim(x->∞) [1-F(x)]/ x^(-a) (forms of 0/0)

    =lim(x->∞) [1-F(x)]' / [(-a)x^(-a-1)] (by L' rule)

    = lim(x->∞) -f(x)/[ (-a)x^(-a-1)]

    = lim(x->∞) x^(a+1) f(x) / a

    hence A=lim(x-∞) x^(a)[1-F(x)] = lim(x->∞) x^(a+1) f(x) / a

    or lim(x->∞) x^a [1-F(x)] = A iff lim(x->∞) x^(a+1) f(x) = aA

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