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416-1sol - Denker SPRING 2010 416 Stochastic Modeling...

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Denker SPRING 2010 416 Stochastic Modeling - Assignment 1 SOLUTIONS Problem 1: (Problem 51, page 92) A coin, having probability p for landing heads, is flipped until head appears for the r -th time. Let N denote the number of flips required. Calculate E [ N ]. Hint: Write N as a sum of geometric random variables. Solution: Let X j denote the waiting time for the j -th head to show up. We claim that X j is a geometric random variable with parameter p . Indeed, if head appears for the j 1-th time, then flipping the coin is just like starting to flip a coin, so the probability of appearing head first in the n -th trial is the same as the probability of having a head the first time in the n -th trial after the j 1-th appearance of head. So, N = r i =1 X i is the waiting time for the r -th head to show up and EN = r summationdisplay i =1 EX i = rEX 1 = r p is the expected time (see book page 68/69 for the expectation of a geometric random variable with parameter p ). Problem 2: (Problem 56, page 93) There are n types of coupons. Each newly obtained coupon is, independently, type i with probability p i , i = 1 , ..., n . Find the expected num-
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