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STAT_333_Assignment_2 - Annotated

STAT_333_Assignment_2 - Annotated - STAT 333 Assignment 2...

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STAT 333 Assignment 2 Due: Friday, March 5 at the beginning of class 1. Suppose we toss a fair coin repeatedly. Let λ be the event “H H T T ”. a. Why is λ a renewal event? b. Use the renewal sequence { r n } to show that λ is recurrent. c. Determine ( ) R s λ (the generating function of { r n }), use it to obtain and prove recurrence. ( ) F s λ d. Use to calculate E[ T ( ) F s λ λ ]. Is λ positive recurrent or null recurrent? e. Expand in a power series and find f ( ) F s λ 8 , the probability that “H H T T” first occurs on trial 8. Give a logical explanation for this probability. f. Use the Renewal Theorem to find E[ T λ ]. Does this agree with the result in d)? 2. Suppose λ is a delayed renewal event. a. Prove the Delayed Renewal Relation: ( ) ( ) ( ) D s F s R s λ λ λ = . Hint: the proof is very similar to the proof of the Renewal Relation in class. Justify your steps carefully. b. Suppose we roll a fair six-sided die repeatedly. Let λ be the event “3 6 3”.
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