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# hw6 - 1 4 Convergence in probability and covergence in L p...

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Homework 6 Probability Theory (MATH 235A, Fall 2007) 1. Discrete independent random variables (a) Show that if X and Y are independent, integer valued random variables, then P ( X + Y = n ) = X k P ( X = k ) P ( Y = n - k ) . (b) Let X and Y be independent Poisson random variables with parameters λ and μ respectively. Prove that X + Y is a Poisson random variable with parameter λ + μ . (c) Let X and Y be independent Binomial random variables with parame- ters ( n,p ) and ( m,p ) respectively. Prove that X + Y is a Binomial random variable with parameter ( n + m,p ). 2. Uncorrelated but not independent random variables Consider the probability space Ω = [ - 1 / 2 , 1 / 2] with the normalized Lebesgue measure on it. Construct two random variables X , Y on Ω that are uncorrelated but not independent. (Hint: consider X ( x ) = x , Y ( x ) = ax 2 + b ). 3. Product of independent random variables Let X,Y > 0 be inde- pendent random variables with distribution functions F and G . (a) Find the distribution function of XY . (b) Compute the distribution function of XY if X and Y are independent random variables uniformly distributed on [0
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Unformatted text preview: , 1]. 4. Convergence in probability and covergence in L p We have shown in class that convergence in L p implies convergence in probability for all p > 0. Show that the converse does not hold. (Prove this by example for p = 1). 5. The coupon collector problem (a) How many times does one need to toss a coin (on average) until the ﬁrst head occurs? (b) How many times does one need to roll a dice (on average) until the ﬁrst “six dots” occurs? (c) How many times does one need to roll a dice (on average) until all faces have appeared at least once? (d) Each time one buys a bag of cheese doodles there is one bonus coupon inside. There are n diﬀerent coupons that are equally likely to be inside any bag. Prove that one needs to buy about Cn log n bags on average in order to have a complete collection of the coupons. (Here C is some constant)....
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