Poisson Distribution

PMF and CDF for rare-event counts

X ~ Poisson(λ = 3) · P(X = 2)

P(X = k) — exactly k

0.224042

P(X = k)

0.224042

P(X ≤ k)

0.42319

P(X ≥ k)

0.800852

Mean (λ)

Variance (λ)

Std. deviation (√λ)

1.7321

Most likely count

About this calculator

The Poisson Distribution Calculator gives the probability of a count of rare, independent events given an average rate λ. It returns the exact probability P(X = k), the cumulative tails P(X ≤ k) and P(X ≥ k), plus the mean, variance, standard deviation, and most likely count — and draws the probability bars with your selection highlighted. Useful for arrivals, defects, calls, decay counts, and other rare-event modeling.

Common examples

λ = 3 arrivals/hour: P(X = 2) = 0.2240
Cumulative: P(X ≤ 2) = 0.4232 when λ = 3
Right tail: P(X ≥ 5) = 0.1847 when λ = 3
No events: P(X = 0) = e^−λ = 0.0498 when λ = 3
Mean λ = 10 ⇒ variance 10, σ = 3.1623, most likely count 10