List of basic formulas related to the Poisson distribution:
- Probability formula:
[
P(X = k) = \frac{\lambda^ke^{-\lambda}}{k!}
]
where:
- ( P(X = k) ) is the probability that the event will occur ( k ) times,
- ( \lambda ) — the average number of events over a fixed interval of time or space,
- ( e ) is the base of the natural logarithm (approximately 2.71828),
- ( k! ) is the factorial of the number ( k ).
- Expected value (average):
[
E(X) = \lambda
] - Variance:
[
Var(X) = \lambda
] - Cumulative distribution function:
[
P(X \leq k) = \sum_{i=0}^{k} P(X = i) = \sum_{i=0}^{k} \frac{\lambda^ie^ {-\lambda}}{i!}
]