STAT 333 Assignment 2

# STAT 333 Assignment 2 - STAT 333 Assignment 2 Due Friday...

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STAT 333 Assignment 2 Due: Friday, March 2 at the beginning of class (or up to 10 minutes in) 1. Consider a sequence of independent tosses of a fair coin. Each toss results in a head H or a tail T. Let λ be the event that the total number of H equals exactly one-third the number of tosses, i.e. we say λ occurs on the n th toss if the total number of H up to and including the n th toss equals n/3. a. Give an argument that λ is a renewal event b. What is the period of λ ? c. Prove that lambda is transient. Why does this make logical sense when you consider the law of large numbers? (that with a large number of trials, the percentage of successes will approach the probability of a success) 2. Suppose a sequence of independent trials X 1 ,X 2 , . . . is generated by randomly picking digits (with replacement) from the set {0, 1, 2, . . ., 9}. Prove that the event “2 3 5” is recurrent by: a. Finding r n = P(“2 3 5” occurs on trial n) and showing that Σr n = ∞ b. Obtaining the generating function R 235 (s) of the renewal sequence {r n }, finding the pgf F 235 (s) of the first waiting time T 235 and evaluating F

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STAT 333 Assignment 2 - STAT 333 Assignment 2 Due Friday...

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