6.262 Discrete Stochastic Processes, Problem Set 6 Solutions
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The purpose of this exercise is to show that, for an arbitrary renewal process, N(t), the number of renewals in (0, t], has finite expectation. a) Let the inter-renewal intervals have the distribution FX (x), with, as usual, FX (0) = 0. Using whatever combination of mathematics and common sense is comfortable for you, show that numbers � > 0 and δ > 0 must exist such that FX (δ) ≤ 1 − �. In other words, you are to show that a positive rv must take on some range of of positive values with positive probability. Solution: For any � < 1, we can look at the sequence of events {X ≥ 1/k; k ≥ 1}. The union of these events is the event {X > 0}. Since Pr{X ≤ 0} = 0, Pr{X > 0} = 1. The events {X ≥ 1/k} are nested in k, so that, from (??),
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