Poisson distribution and its typical use cases.

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Answer

$X ~ Poisson(λ): count of events in a fixed interval / area, when events happen at constant rate λ independently.$
$P(X = k)\; = \;{\lambda}^{k}\; \cdot \;e$
$E[X] = Var(X) = λ.$
$Use cases: website clicks per minute, defects per page, insurance claims per year.$
$Limit of Binomial(n, p) as n \to \infty, p \to 0, np \to λ.$
$Assumes memorylessness — real data with time-varying rate (bursty traffic) needs negative binomial or NHPP.$

Check yourself — multiple choice

E ≠ Var
Count events in fixed interval; E[X] = Var(X) = λ; limit of Binomial when np → λ; watch for over-dispersion in real data
Only continuous
Var = 1

Poisson: E = Var = λ; use for event counts with constant rate.

#distributions#probability

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