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Poisson Distribution Calculator

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Poisson Distribution Calculator

The Poisson distribution gives the probability of observing k events in a fixed interval when events occur independently at a constant average rate λ: calls to a helpdesk per hour, typos per page, goals per match. Enter λ, choose the probability type and k, and the calculator applies P(X = k) = e⁻ᵏλᵏ ÷ k! and sums the terms you need.

Alongside the answer you get the single-term computation with numbers, the cumulative rule, the mean and variance (both equal to λ), a bar chart with the selected range highlighted and a table of probabilities for k from 0 upward. Because λ can be scaled, a rate of 2 per hour becomes λ = 6 for a three-hour window.

The Poisson also approximates the binomial distribution when n is large and p is small (λ = np), and the time between Poisson events follows the exponential distribution. This page matches poissonpdf and poissoncdf on the TI-84 and POISSON.DIST in Excel.

Result and step-by-step solution

P(X ≤ 2)
0.32085

Mean = variance = λ = 3.5, SD = 1.8708

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  1. Formula
    P(X = k) = e−λ × λk ÷ k!
  2. Single term at k = 2
    P(X = 2) = e−3.5 × 3.52 ÷ 2! = 0.030197 × 12.25 ÷ 2 = 0.18496
  3. Add the terms in the range
    P(X ≤ 2) = 0.32085
  4. Probability table
    kP(X = k)P(X ≤ k)P(X ≥ k)
    00.03020.03021
    10.105690.135890.9698
    20.184960.320850.86411
    30.215790.536630.67915
    40.188810.725440.46337
    50.132170.857610.27456
    60.07710.934710.14239
    70.038550.973260.06529
    80.016870.990130.02674
    90.006560.996690.00987
    100.00230.998980.00331
    110.000730.999710.00102
    120.000210.999920.00029
    130.000060.999980.00008

Frequently asked questions

What is λ in the Poisson distribution?
The average number of events in the interval you are asking about. It is both the mean and the variance of the distribution.
How do I find the probability of at least one event?
P(X ≥ 1) = 1 − P(X = 0) = 1 − e⁻ᵏ. With λ = 3.5 that is 1 − 0.0302 = 0.9698.
Can λ be a decimal?
Yes. λ is an average, so 3.5 or 0.25 are valid; only k must be a whole number.
When is Poisson a good approximation to the binomial?
When n is large (say 50+) and p is small (0.1 or less), with λ = np. The approximation improves as n grows and p shrinks.

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Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

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