{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

STA3007_0506_t08

STA3007_0506_t08 - STA 3007 Applied Probability Tutorial 8...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
ii. Let { X ( t ); t 0 }
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}