Lec9[1] - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 9: Waiting for patterns Lecturer: Jim Pitman 1 Waiting for patterns Expected waiting time for patterns in Bernoulli trials Suppose X 1 ,X 2 ,... are independent coin tosses with P ( X i = H ) = p , P ( X i = T ) = 1 - p = q . Take a particular pattern of some finite length K , say HH . ..H | {z } K or HH . | {z } K - 1 T . Let T pat := first n s.t. ( X n - K +1 n - K +2 ,...,X n | {z } K ) = pattern You can try to find the distribution of T pat , but you will find it very difficult. We can compute E [ T pat ] with our tools. Start with the case of pat = HH . | {z } K . For K = 1, P ( T H = n ) = q n - 1 p, n = 1 , 2 , 3 ... E ( T H ) = X n nq n - 1 p = 1 p For a general K , notice that when the first T comes (e.g. HHHT , considering some K 4), we start the counting process again with no advantage. Let m K be the expected number of steps to get K heads in a row. Let T T be the time of the first tail. Observe : If T T > K , then T HH. ..H = K ; if T T = j K , then T ..H = j + (fresh copy of) T ..H .
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Lec9[1] - Stat 150 Stochastic Processes Spring 2009 Lecture...

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