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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 '07
 RomanVershynin
 Probability, Probability theory, CN, independent random variables

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