# hw7 - Homework Set No 7 Probability Theory(235A Fall 2011...

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Homework Set No. 7 – Probability Theory (235A), Fall 2011 Due: 11/15/11 1. Prove that if F and ( F n ) n =1 are distribution functions, F is continuous, and F n ( t ) F ( t ) as n → ∞ for any t R , then the convergence is uniform in t . 2. Let ϕ ( x ) = (2 π ) - 1 / 2 e - x 2 / 2 be the standard normal density function. (a) If X 1 ,X 2 ,... are i.i.d. Poisson(1) random variables and S n = n k =1 X k (so S n Poisson( n )), show that if n is large and k is an integer such that k n + x n then P ( S n = k ) 1 n ϕ ( x ) . Hint: Use the fact that log(1 + u ) = u - u 2 / 2 + O ( u 3 ) as u 0. (b) Find lim n →∞ e - n n k =0 n k k ! . (c) If X 1 ,X 2 ,... are i.i.d. Exp(1) random variables and denote S n = n k =1 X k (so S n Gamma( n, 1)), ˆ S n = ( S n - n ) / n . Show that if n is large and x R is ﬁxed then the density of ˆ S n satisﬁes f ˆ S n ( x ) ϕ ( x ) . 3. (a)

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## This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.

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hw7 - Homework Set No 7 Probability Theory(235A Fall 2011...

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