STA3007_0506_t08

STA3007_0506_t08 - STA 3007 Applied Probability 2005...

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Unformatted text preview: STA 3007 Applied Probability 2005 Tutorial 8 1. Poisson Process (a) Poisson Process Definition A Poisson process of intensity, or rate, λ > is an integer-valued stochastic pro- cess { X ( t ); t ≥ } for which (1) for any time points t = 0 < t 1 < t 2 < ·· · < t n , the process increments X ( t 1 )- X ( t ) ,X ( t 2 )- X ( t 1 ) ,...,X ( t n )- X ( t n- 1 ) are independent random variables; (2) for s ≥ and t > , the random variable X ( s + t )- X ( s ) has the Poisson distribution Pr { X ( s + t )- X ( s ) = k } = ( λt ) k e- λt k ! for k = 0 , 1 ,... ; (3) X(0)=0. In particular, observe that if X ( t ) is a Poisson process of rate λ > , then the moments are E [ X ( t )] = λt and Var[ X ( t )] = λt. For λ > is a constant value, it is called a homogenous Poisson Process. For λ = λ ( t ) > is a function of time, it is called a nonhomogenous Poisson Process. Example: i. Suppose that customers arrive at a facility according to a Poisson process having rate λ = 2 . Let X ( t ) be the number of customers that have arrived up to time t . Determine the following probabilities and conditional probabilities: (a) Pr...
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This note was uploaded on 05/21/2011 for the course STA 3007 taught by Professor Kb during the Spring '11 term at CUHK.

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STA3007_0506_t08 - STA 3007 Applied Probability 2005...

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