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Poisson Distribution Calculator

Enter the average number of events per interval (λ) and the count k you are interested in.

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Formula

  • P(X = k) = e^(−λ) × λ^k ÷ k!
  • P(X ≤ k) is the sum of P(X = i) for i from 0 to k
  • P(X ≥ k) = 1 − P(X ≤ k − 1)
  • Mean and variance are both λ

Worked example

λ = 4, k = 2:

  • P(X = 2) = e^(−4) × 16 ÷ 2 = 0.1465
  • P(X ≤ 2) = 0.2381
  • P(X ≥ 2) = 0.9084
  • Mean = variance = 4

Frequently asked questions

What is the Poisson probability formula?

P(X = k) = e^(−λ) × λ^k ÷ k!. With λ = 4 and k = 2 that is e^(−4) × 16 ÷ 2 = 0.1465.

What are the mean and variance of a Poisson distribution?

Both equal λ. If a call centre gets 4 calls a minute on average, the variance is 4 and the standard deviation is 2.

How do I find the chance of at least k events?

Use P(X ≥ k) = 1 − P(X ≤ k − 1). With λ = 4 and k = 2, P(X ≤ 1) = 0.0916, so P(X ≥ 2) = 0.9084.

When does the Poisson distribution apply?

For counts of independent events that occur at a constant average rate in a fixed interval of time or space, such as arrivals, defects or typos per page.

How we calculate

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