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5.1.4 Customers arrive at a service facility according to a Poisson process of rate customer/hour. Let X(t) be the number of customers that have arrived up to time t. (a) What is Pr{X(t)=k} for k = 0, 1,...? (b) Consider fixed times 0

5.1.4 Customers arrive at a service facility according to a Poisson process of rate customer/hour. Let X(t) be the number of customers that have arrived up to time t. (a) What is Pr{X(t)=k} for k = 0, 1,...? (b) Consider fixed times 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 78E
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5.1.4 Customers arrive at a service facility according to a Poisson process of rate customer/hour. Let X(t) be the number of customers that have arrived up to time t. (a) What is Pr{X(t)=k} for k = 0, 1,...? (b) Consider fixed times 0<s<t. Determine the conditional probability Pr{X(t)=n+k|X(s) = n} and the expected value E[X(t)X(s)].
Transcribed Image Text:5.1.4 Customers arrive at a service facility according to a Poisson process of rate customer/hour. Let X(t) be the number of customers that have arrived up to time t. (a) What is Pr{X(t)=k} for k = 0, 1,...? (b) Consider fixed times 0<s<t. Determine the conditional probability Pr{X(t)=n+k|X(s) = n} and the expected value E[X(t)X(s)].
5.1.4 (a) (at)ke-λt k! k = 0, 1, ...; (b) Pr{X(t)=n+k\X(s) = n} = [(t−s)]ke¯(-s) E[X(t)X(s)]=²ts +λs. k!
Transcribed Image Text:5.1.4 (a) (at)ke-λt k! k = 0, 1, ...; (b) Pr{X(t)=n+k\X(s) = n} = [(t−s)]ke¯(-s) E[X(t)X(s)]=²ts +λs. k!
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