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STA3007_0506_t08

# STA3007_0506_t08 - STA 3007 Applied Probability Tutorial 8...

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STA 3007 Applied Probability 2005 Tutorial 8 1. Poisson Process (a) Poisson Process Definition A Poisson process of intensity, or rate, λ > 0 is an integer-valued stochastic pro- cess { X ( t ); t 0 } for which (1) for any time points t 0 = 0 < t 1 < t 2 < · · · < t n , the process increments X ( t 1 ) - X ( t 0 ) , X ( t 2 ) - X ( t 1 ) , . . . , X ( t n ) - X ( t n - 1 ) are independent random variables; (2) for s 0 and t > 0 , 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 > 0 , then the moments are E [ X ( t )] = λt and Var[ X ( t )] = λt. For λ > 0 is a constant value, it is called a homogenous Poisson Process. For λ = λ ( t ) > 0 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 { X (1) = 2 } (b) Pr { X (1) = 2 and X (3) = 6 } (c) Pr { X (1) = 2 | X (3) = 6 } (d) Pr { X (3) = 6 | X (1) = 2 } 1

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ii. Let { X ( t ); t 0 }
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