Poisson Distribution Calculator

Your details

The average number of events expected in one interval (must be >= 0).
events
Scale lambda to cover multiple intervals. If lambda = 3/hour and you want 2 hours, enter 2 here. Effective lambda = lambda x intervals.
intervals
Exact mode computes point and cumulative tail probabilities. Range mode computes P(x1 <= X <= x2) and the complementary outside-range probability.
The exact count you want the probability of. Must be a whole number >= 0.
events
P(X = k)
14.6525%

Probability of exactly k events.

Effective lambda (scaled)4
P(X <= k)23.8103%
P(X < k)9.1578%
P(X >= k)90.8422%
P(X > k)76.1897%
Mean (lambda)4
Variance4
Standard deviation2
P(X = k)14.6525%
P(X <= k)23.8103%
P(X >= k)90.8422%

With an effective mean of 4, the chance of exactly 2 events is 14.6525%.

  • The most likely single outcome is near k = 4 or k = 4, where the PMF peaks.
  • There is a 23.81% chance of 2 or fewer events, and a 90.842% chance of 2 or more.
  • The standard deviation is 2, and because Poisson variance equals its mean, the spread grows as lambda rises.

Next stepUse the range mode to compute the probability of a band of outcomes, or increase the interval count to scale the rate forward in time.

 = Powered by OnlyCalculators