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Poisson Distribution Percent Point

Enter a probability p and a mean lambda to find the smallest integer k whose cumulative probability is at least p, with the actual CDF, probability mass, mean and variance.

Input

Enter a probability p and a mean lambda to find the smallest integer k whose cumulative probability is at least p (the lower percent point).

A value greater than 0 and less than 1 (e.g. 0.95)

The mean number of events, greater than 0 (e.g. 4)

Result

Smallest k with P of X at most k at least 0.95 (lambda = 4)

8

Actual CDF P of X at most k

0.97863657

Mass P of X equals k

0.02977018

Upper probability P of X at least k

0.05113362

Mean

4

Variance

4

Probability mass P of X equals k

01234567891011

How it works

  • The Poisson probability mass is P of X equals k = e to the power minus lambda times lambda to the power k divided by k factorial.
  • The percent point is the smallest integer k whose cumulative probability P of X at most k is at least the given probability p.
  • Both the mean and the variance of a Poisson distribution equal lambda.
  • The cumulative probability is computed with the regularized incomplete gamma function.

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