# Difference between revisions of "Stochastic Processes"

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(Created page with "Stochastic Process as taught in Durett's Book in STAT650. ==Markov Chains== ==Poisson Processes== ==Renewal Processes== ==Queueing Theory==") |
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==Markov Chains== | ==Markov Chains== | ||

==Poisson Processes== | ==Poisson Processes== | ||

+ | The following 3 definitions of the Poisson process are equivalent. | ||

+ | * A Poisson process with rate <math>\lambda</math> is a counting process where the number of arrivals <math>N(s+t)-N(s)</math> in a given time period <math>t</math> has distribution <math>Poisson(\lambda t)</math> | ||

+ | * A Poisson process is a renewal process with rate <math>1/\lambda</math> | ||

+ | * A Poisson process is a continuous time markov chain with <math>P(N(h)=1) = \lambda h + o(h)</math> and <math>P(N(h) \geq 2) = o(h)</math> | ||

+ | |||

==Renewal Processes== | ==Renewal Processes== | ||

==Queueing Theory== | ==Queueing Theory== |

## Latest revision as of 13:44, 11 November 2019

Stochastic Process as taught in Durett's Book in STAT650.

## Markov Chains

## Poisson Processes

The following 3 definitions of the Poisson process are equivalent.

- A Poisson process with rate \(\displaystyle \lambda\) is a counting process where the number of arrivals \(\displaystyle N(s+t)-N(s)\) in a given time period \(\displaystyle t\) has distribution \(\displaystyle Poisson(\lambda t)\)
- A Poisson process is a renewal process with rate \(\displaystyle 1/\lambda\)
- A Poisson process is a continuous time markov chain with \(\displaystyle P(N(h)=1) = \lambda h + o(h)\) and \(\displaystyle P(N(h) \geq 2) = o(h)\)