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# lec0412 - IEOR 4106 Spring 2011 Professor Whitt Topics for...

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IEOR 4106, Spring 2011, Professor Whitt Topics for Discussion: Tuesday, April 12 Renewal Theory: Patterns 1. Patterns: see § 3.6.4 and § 7.9 Consider successive independent flips of a biased coin. On each flip, the coin comes up heads (H) with probability p or tails (T) with probability q = 1 - p , where 0 < p < 1. A given segment of finitely many consecutive outcomes is called a pattern . The pattern is said to occur at flip n if the pattern is completed at flip n . For example, the pattern A HTHTHT occurs at flips 8 and 10 in the sequence TTHTHTHTHTTTTHHHT . . . and at no other times among the first 17 flips. WARMUP For parts (a) and (b) below, assume that p = 1 / 2, but for later parts do not make that assumption. (a) Which pattern occurs more frequently in the long run: A HHH or B HTH ? (b) For patterns A and B in part (a), let N A and N B be the numbers of flips until the patterns A and B , respectively, first occur. Is E [ N A ] = E [ N B ]? MAIN PROBLEM Now we revert to general probabilities p and q = 1 - p . (c) What is the probability that pattern A HTHTHT occurs at flip 72? (d) Suppose that pattern A from part (c) does indeed occur at flip 72. What is the expected number of flips until pattern A occurs again? (e) Let N A ( n ) be the number of occurrences of pattern A in the first n flips, where A is again the pattern in part (c). Does N A ( n ) n x as n → ∞ w.p.1? If so, what is the limit x ? (f) What is E [ N A ], the expected number of flips until pattern A HTHTHT first occurs? (g) What is the probability that pattern A occurs before pattern B TTH ? That is, what is P ( N A < N B )? NOTATION (i) A or B , a possible pattern, as in parts (a) and (c) above (ii) N A , a random variable, equal to the number of flips until the pattern A occurs for the first time, as defined in part (b) above (iii) A ( n ), the event that pattern A occurs (ends) on flip n . Note that in part (c), we are asking for the value of P ( A (72)). (iv) T A , a random variable, equal to the number of flips until the pattern A occurs again, after it has occurred previously. This formalized the question in part (d).

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(v) N A ( n ), a random variable, equal to the number of times that pattern A occurs in the first n flips, as defined above in part (e) (vi) N A B , a random variable, equal to the number of flips until the pattern B first occurs starting from the occurrence of A . This is new notation used to derive the answers to parts (f) and (g).
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