7. Below X1, X2,... are assumed to be independent random variables. Use The- orem 3.25 to prove the following results. (

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7 Below X1 X2 Are Assumed To Be Independent Random Variables Use The Orem 3 25 To Prove The Following Results 1 (84.11 KiB) Viewed 20 times

7. Below X1, X2,... are assumed to be independent random variables. Use The- orem 3.25 to prove the following results. (a) If Xn - Bin(n, 1/n), then, as n → 00, X, converges in distribution to a Poi(2) random variable. (b If X, - Geom(1/n), then, as n , x, converges in distribution to an Exp(2) random variable. (©) If X, ~ U(0, 1) and Mn = max(X1, X2, ..., X.), then, as n + 0, n(1 - M.) converges in distribution to an Exp(1) random variable. Can you also find a proof without using characteristic functions? Theorem 3.25: Characteristic Function and Convergence in Distribution Suppose that wx, (O).V x2(t),... are the characteristic functions of the se- quence of random vectors X1, X2, ... andự x(t) is the characteristic function of X. Then, the following three statements are equivalent: 1. limo–ox (t) = xx (1) for all t € R". 2. X, X 3. lim - Eh(x) = E(X) for all bounded continuous functions h : R! → R. +00