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:

- The probability distribution of
*N*(*t*) is a Poisson distribution. - The probability distribution of the waiting time until the next occurrence is an exponential distribution.
- The occurrences are distributed uniformly on any interval of time. (Note that
*N*(*t*), the total number of occurrences, has a Poisson distribution over (0,*t*], whereas the location of an individual occurrence on*t*∈ (*a*,*b*] is uniform.)

[/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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