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WaitingTimeVariables_R

# WaitingTimeVariables_R - • Proper random variables can be...

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C.D. Cutler STAT 333 Discrete Waiting Time Random Variables Let E be a possible event of a stochastic process X 1 , X 2 , X 3 , . . . . Let X = waiting time (number of steps or trials) until E first occurs in the sequence. The potential range of X is { 1 , 2 , 3 , 4 , . . . } ∪ {∞} where X = n means E first occurs on trial n and X = means that E never occurs. For each x in the potential range we define f ( x ) = P ( X = x ). This is the probability mass function (p.m.f.) of X . We must include the value f ( ) = P ( E never occurs). X is called proper if f ( ) = 0 and improper if f ( ) > 0. Proper random variables are those with which you are familiar from other courses; they satisfy x =1 f ( x ) = 1 (there is no probability at and their range can be restricted to the integers). Proper random variables can be further classified into two types:
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Unformatted text preview: • Proper random variables can be further classiFed into two types: Short proper waiting times: E ( X ) < ∞ Null (long) waiting times: E ( X ) = ∞ Infnite Sums and Infnite Products: By deFnition ∞ s k =1 a k = lim n →∞ n s k =1 a k and ∞ p k =1 a k = lim n →∞ n p k =1 a k The Sum-Product Lemma: Suppose 0 < a n < 1 for each n . Then ∞ p k =1 (1-a k ) > if and only if ∞ s k =1 a k < ∞ A Lower Bound In The Geometric Case: Suppose 0 < p < 1 / 2 and a k = p k . Then ∞ p k =1 (1-p k ) ≥ 1-∞ s k =1 p k = 1-p 1-p = 1-2 p 1-p...
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