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hw8 - Homework Set No 8 Probability Theory(235A Fall 2009...

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Homework Set No. 8 – Probability Theory (235A), Fall 2009 Posted: 11/17/09 — Due: 11/24/09 1. (a) Prove that if X, ( X n ) n =1 are random variables such that X n X in probability then X n = X . (b) Prove that if X n = c where c R is a constant, then X n c in probability. (c) Prove that if Z, ( X n ) n =1 , ( Y n ) n =1 are random variables such that X n = Z and X n - Y n 0 in probability, then Y n = Z . 2. (a) Let X, ( X n ) n =1 be integer-valued r.v.’s. Show that X n = X if and only if P ( X n = k ) P ( X = k ) for any k Z . (b) If λ > 0 is a fixed number, and for each n , Z n is a r.v. with distribution Binomial( n, λ/n ), show that Z n = Poisson( λ ) . 3. Let f ( x ) = (2 π ) - 1 / 2 e - x 2 / 2 be the density function of the standard normal distribution, and let Φ( x ) = R x -∞ f ( u ) du be its c.d.f. Prove the inequalities 1 x + x - 1 f ( x ) 1 - Φ( x ) 1 x f ( x ) , ( x > 0) . (1) Note that for large x this gives a very accurate two-sided bound for the tail of the normal distribution. In fact, it can be shown that 1 - Φ( x ) = f ( x ) · 1 x + 1 x + 2 x + 3 x + 4 x + ...
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