Where: e ≈ 2.71828, λ = average rate, k = number of events
E[X] = λ, Var(X) = λ
Poisson probability calculates the probability of exactly k events occurring in a fixed interval given the average rate λ.
⚠λ must be non-negative, k must be a non-negative integer.
What is Poisson Probability?
The Poisson distribution models the number of events occurring in a fixed interval of time or space, where these events occur with a known constant average rate and independently of the time since the last event.
Conditions
Rare events, independent occurrences, known average rate
Poisson probability calculates the probability of exactly k events occurring in a fixed interval (time/space), given average rate λ. P(X=k) = (e^(-λ) × λ^k) / k!.
When to use Poisson distribution?▼
Use Poisson for counting events in fixed interval: rare events, independent occurrences, known average rate, events don't affect each other.
What is lambda (λ)?▼
λ is the average rate parameter - average number of events in the interval. For Poisson, E[X] = Var(X) = λ.
Relationship between Poisson and binomial?▼
Poisson approximates binomial when n is large, p is small, and λ = n×p is moderate. Useful for rare events.
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