Poisson Distribution Calculator

The Poisson distribution gives the probability of a number of events happening in a fixed interval when they occur independently at a constant average rate, such as calls per hour or typos per page. This calculator finds the probability of exactly, at most or at least k events and charts the distribution.

Try:

How to use the poisson distribution calculator

  1. Enter the average rate λ.
  2. Enter the number of events k and choose =, ≤ or ≥.
  3. Press Calculate to see the probability and chart.

Formula

P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}

Worked example: λ = 4, exactly 2

  • Average rate λ: 4
  • Number of events k: 2
  • Find: P(X = k)

✓ Answer checked

P(X=2)≈0.14652511P(X = 2) \approx 0.14652511
As a percentage
14.6525%
Mean = variance
λ=4\lambda = 4
Standard deviation
λ≈2\sqrt{\lambda} \approx 2
00.050.10.150: 0.0183156388887341801: 0.0732625555549367312: 0.1465251111098734323: 0.1953668148131645434: 0.1953668148131646245: 0.156293451850531756: 0.1041956345670210267: 0.059540362609726378: 0.02977018130486314589: 0.013231191691050281910: 0.0052924766764201071011: 0.0019245369732436791112: 0.00064151232441455771213: 0.000197388407512171741314: 0.00005639668786062049414
Probability of each number of events when the average is 4. Highlighted bars make up the answer.
Step-by-step working (2 steps)

  1. Poisson formula Poisson distribution

    Use this when events happen independently at a constant average rate λ.

    P(X=i)=e−λλii!P(X = i) = \frac{e^{-\lambda}\lambda^{i}}{i!}

  2. Substitute

    Put λ = 4 and i = 2.

    P(X=2)=e−4 422!≈0.14652511P(X = 2) = \frac{e^{-4}\, 4^{2}}{2!} \approx 0.14652511
Formulas used
P(X=k)=e−λλkk!P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}
How this was checked
  • ✓ All the probabilities add up to 1.
  • ✓ Adding the upper tail directly gives the same answer.

Probabilities rounded to 8 significant figures.

Frequently asked questions

When should I use the Poisson distribution?

When counting independent events over a fixed time, area or volume, with a known average rate and no upper limit on the count.

What are the mean and variance?

Both equal λ, so the standard deviation is √λ.

What is P(X = 2) when λ = 4?

e⁻⁴ × 4² ÷ 2! ≈ 0.1465, or about 14.7%.

Related calculators

Binomial Distribution Calculator

Calculate binomial probabilities for exactly, at most or at least k successes in n trials.

Normal Distribution Calculator

Find normal probabilities P(X < a), P(X > a) or P(a < X < b) from the mean and standard deviation, with z-scores and a shaded bell curve.

Permutation & Combination Calculator

Calculate permutations (nPr) and combinations (nCr) exactly, even for large numbers, plus the counts with repetition.

Related guides