Statistics & Probability

Poisson Distribution Calculator

Find the probability of k events in an interval when events happen at an average rate λ.

By TheCalcTools Updated Formula checked and tested

Your numbers

Your results

P(X = k)
P(X = k)
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P(X ≤ k)
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P(X ≥ k)
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Standard deviation
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Poisson Distribution Calculator by TheCalcTools

How the Poisson Distribution Calculator works

Enter the average number of events per interval, λ, and the count k you want.

The Poisson formula gives the exact probability.

Mean and variance both equal λ.

The formula

P(X = k) = λᵏ e⁻λ ÷ k!Mean = Variance = λ

Worked example

Using the default values shown in the calculator:

Average rate (λ)3
Events (k)2
  1. P(X = 2) = 3² × e⁻³ ÷ 2! = 0.2240.
  2. P(X ≤ 2) = e⁻³ × (1 + 3 + 4.5) = 0.4232.
  3. P(X ≥ 2) = 0.8009.

Result:

P(X = k)0.2240
P(X ≤ k)0.4232
P(X ≥ k)0.8009
Standard deviation1.7321

Frequently asked questions

When is the Poisson distribution used?

For counts of rare, independent events over a fixed time or area, such as calls per hour or defects per metre.

What if the rate changes over time?

Then the Poisson model is not a good fit unless you split the period into parts with similar rates.

Sources and further reading

Binomial Distribution CalculatorCalculate binomial probabilities for n trials with success probability p Normal Distribution CalculatorEnter the mean, standard deviation and a value x to find its z-score and probabilities Erlang B CalculatorHow many calls will be blocked when traffic meets a fixed number of lines? Z-Score CalculatorSee how many standard deviations a value is from the mean Hypergeometric Probability CalculatorDraw n items without replacement from a population containing K successes Geometric Distribution CalculatorHow many attempts until the first success?

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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.