Poisson Process

Poisson process deals with occurrences that happen with low probability, and it  connects possison distribution and exponential distribution nicely. According to wikipedia entry, http://en.wikipedia.org/wiki/Poisson_process:
[quote]

The basic form of Poisson process, often referred to simply as “the Poisson process”, is a continuous-time counting process {N(t), t ≥ 0} that possesses the following properties:

  • N(0) = 0
  • Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other)
  • Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval)
  • No counted occurrences are simultaneous.

Consequences of this definition include:

[/quote]

If lambda is the average of observing something, then N(t) ~ Poisson(t*lambda) while the time between observing something follows Exp(lambda) where Exp represents exponential distribution.

If t = 1, then Possison(lambda) represents the probability of observing something in unit time while Exp(lambda) represents the time between observing something.




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