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. prob It has the probability distribution

Proof

proof

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.

Proof of 1

By ^P1

as required.

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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