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)
3
Variance (λ)
3
Std. deviation (√λ)
1,7321
Most likely count
3
Poisson probability bars with the selected counts highlighted. P(X = 2) = 0,224042Poisson probability bars with the selected counts highlighted. P(X = 2) = 0,224042012345678910

Acerca de esta calculadora

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.

Ejemplos comunes

  • 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