Poisson Mean Calculator (Confidence Limits and Inverse)

Back-solve the Poisson mean lambda from an observed count and a cumulative probability. Supports lower and upper confidence limits and a general inverse, with PMF and cumulative probabilities.

Input

Back-solve the Poisson mean lambda from an observed count k and a cumulative probability p. Choose a lower or upper confidence limit or a general inverse.

Observed count k

Enter an integer of 0 or more.

Cumulative probability p

A value greater than 0 and less than 1. Use 0.025 per tail for a 95 percent two sided interval.

Inverse mode

Select which cumulative probability is matched to p when solving for lambda.

Result

Mean lambda for k = 3 and p = 0.95

1.3663184

Upper confidence limit P(X less or equal k) = p

Lower cumulative P(X less or equal k)

0.95

Upper cumulative P(X greater or equal k)

0.15842263

Probability mass P(X = k)

0.10842263

Mean

1.3663184

Variance

1.3663184

Probability mass distribution P(X = i)

How it works

The Poisson probability mass is P(X = i) = e^(-lambda) lambda^i / i!. Both the mean and the variance equal lambda.
The lower cumulative P(X less or equal k) equals the regularized upper incomplete gamma Q(k+1, lambda), and the upper cumulative P(X greater or equal k) equals the regularized lower incomplete gamma P(k, lambda).
The lower limit mode solves P(X greater or equal k) = p for lambda, the upper limit mode solves P(X less or equal k) = p for lambda, both by bisection. The general inverse mode solves P(X less or equal k) = p for lambda.
For a 95 percent two sided interval, assign probability 0.025 to each tail. Use the lower mode with p = 0.025 for the lower limit and the upper mode with p = 0.025 for the upper limit.
Enter the observed count k as an integer of 0 or more, and the cumulative probability p as a value greater than 0 and less than 1.

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