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), then, as n - co, X, converges in distribution to a Poi(1) random variable. (b) If X, ~ Geom(1), then, as n - co, -X,, converges in distribution to an Exp(1) random variable. (c) If X, ~ U(0, 1) and M, = max(X1, X2, . .., X,, ), then, as n -> co, 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, (t), wx, (t), . .. are the characteristic functions of the se- quence of random vectors X1, X2, ... and wx(t) is the characteristic function of X. Then, the following three statements are equivalent: 1. limn-+0Vx.(t) = yx(t) for all t e R". 2. Xn LX. 3. limn-+& Eh(X,) = Eh(X) for all bounded continuous functions h : Rd - R