Poisson Distribution Calculator

From the mean lambda and a count k, compute the Poisson probability mass P(X=k), cumulative P(X less than or equal to k), and upper tail P(X at least k), with a bar chart of the distribution.

Input

Enter the mean lambda and a count k to compute the Poisson probability mass P(X=k), the cumulative probability, and the upper probability.

Count k (non-negative integer)

Enter the number of occurrences whose probability you want.

Mean lambda (greater than 0)

Enter the average number of occurrences per interval (the expected value).

Result

P(X = 3) with lambda = 4

0.19536681

P(X less than or equal to k)

0.43347012

P(X greater than or equal to k)

0.76189669

Mean

Variance

Probability mass P(X = k) distribution

How it works

The Poisson distribution is a discrete distribution giving the probability of exactly k events when events occur on average lambda times in a fixed interval.
The probability mass function is P(X=k) = e^(-lambda) lambda^k / k factorial. Enter k as a non-negative integer and lambda as a positive value.
The cumulative probability P(X less than or equal to k) is the sum of the probability masses from i=0 to k.
The upper probability P(X at least k) is computed as 1 minus P(X less than or equal to k-1).
For a Poisson distribution the mean and the variance are both equal to lambda.
Probabilities are computed in the logarithmic domain (with the log factorial via the gamma function) so large lambda or k do not overflow.

Reviews

Tell us what you think of this calculator.