[機統] 關於機率論T分配的問題!
(a) Suppose that Xn~ tn follows a t-distribution with n degrees of freedom.
Let fn(x) be the density of X. Find lim n->∞ fn(x).
What does this tell you about the distribution of Xn for large n? Hint:
Stirling’s approximation implies that
lim n->∞ Γ(n)．√n/2π．(e/n)^n=1.
You may this useful for computing the desired limit.
(b) Suppose that X1;X2; ::: ~ uniform(0; 1) are independent,so that
P(X1< =x1; :::;Xn< =xn) =P(X1 < = x1).....P(Xn < = xn).
Let Vn = n[1 - max{X1; ::;Xn}.
Compute lim n->∞ Pn(Vn < = x)
for x > 0.
What does this tell you about the distribution of Vn when n is large?
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