Poisson Distribution Calculator
Compute Poisson probabilities for a given rate (lambda) and number of events.
The expected number of events in the interval (must be greater than 0).
A non-negative integer (0, 1, 2, …).
P(X = k)
P(X ≤ k) cumulative
P(X ≥ k)
P(X < k)
Calculation Steps
What the Poisson Distribution Measures
The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when these events happen with a known constant mean rate and independently of the time since the last event. It is widely used for counting rare events, such as calls arriving at a call center, decay events from a radioactive source, or typos on a page.
The Formula
The probability of observing exactly k events when the mean rate is λ is given by:
- P(X = k): (λ^k · e^(−λ)) / k!
- P(X ≤ k): Σ from i = 0 to k of P(X = i)
Mean and Variance
A distinctive property of the Poisson distribution is that both its mean and its variance equal λ. The parameter λ must be positive, while k must be a non-negative integer. As λ grows large, the Poisson distribution approaches a normal distribution.