Discrete random variable

Poisson distribution

The Poisson distribution describes the probability of events occurring in a fixed time interval, where is the expected number of occurrences and the time of each event is independent from any prior ones. #m/def/prob It has the probability distribution

Properties

1. Expectation, variance:
2. Moment-generating function:
3. Probability generating function:

Additionally

1. If and are independently distributed then
2. If and where are independently distributed then and .
3. Conversely, if and then and are independently distributed.

Poisson paradigm

Let be independent events with small. Then is approximated by where , with

for any . See also Poisson process.

Relationship to other distributions

• If and are independently distributed, then the conditional distribution of given is .
• As and as remains fixed .
• By the Central limits theorem as

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