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Unformatted text preview: Stat 333 Winter 2009 Assignment #2 Due Tuesday, February 24 during the midterm. The assignment is complete. Posted Tuesday, February 3: 1. Suppose we have a sequence of independent S or F trials where the probability of S on the n th trial, n = 1 , 2 , 3 ,... , is p n = 4 n . Let Y = number of trials required to obtain the first S . Find a lower bound on the probability P ( Y = ∞ ) to show that Y is an improper waiting time variable. Also indicate how you could use the SumProduct Lemma to show that Y is improper. 2. Below are some hypothetical first waitingtime distributions f k = P ( T λ = k ) where λ is a renewal event of a stochastic process. In each case determine whether λ is transient or recurrent; if recurrent, further determine whether λ is positive recurrent or null recurrent by computing E ( T λ ). (a) P ( T λ = k ) = 3 k k = 1 , 2 , 3 ,... (b) P ( T λ = k ) = 2 k k = 1 , 2 , 3 ,... (c) P ( T λ = k ) = 6 π 2 k 2 k = 1 , 2 , 3 ,... Note: ∑ ∞ k =1 k 2 = π 2 / 6. 3. (a) Find the interval of convergence of the following power series: ∞ summationdisplay n =0 ( 1) n 2 n +2 s n = 4 8 s + 16 s 2 32 s 3 + 64 s 4 ... and determine its functional form A ( s ). (b) Expand the function f ( s ) = e s / (1 6 s ) in a power series, determining the first 5 coefficients a ,a 1 ,a 2 ,a 3 ,a 4 . What is the interval of convergence? (Hint: use known power series rather than proceeding from a Taylor series expansion.) 4. Suppose λ is a renewal event. Let V λ = total number of occurrences (visits) of λ , and suppose that E ( V λ ) = 10....
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 Spring '08
 Chisholm
 Power Series, Probability, Probability theory, Recurrence relation, probability generating function

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