Poisson distribution calculator
Compute the Poisson probability P(X = k), the cumulative P(X ≤ k), the upper tail P(X > k), the mean, variance and mode, with a distribution table and a binomial approximation comparison.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
Result
0.2240420.4231900.576810331.7320513| k | P(X = k) | P(X ≤ k) |
|---|---|---|
| 0 | 0.049787 | 0.049787 |
| 1 | 0.149361 | 0.199148 |
| 2 | 0.224042 | 0.423190 |
| 3 | 0.224042 | 0.647232 |
| 4 | 0.168031 | 0.815263 |
| 5 | 0.100819 | 0.916082 |
With λ = n·p the binomial distribution approaches the Poisson distribution as n grows and p shrinks.
30.2240420.2251530.001111What this tool does
- Queueing and arrivals: with λ = 3 calls per hour you can read off the chance of exactly 2 calls or of at most 3.
- Quality control and count data: defects per unit area, particles per minute or page views per minute are all rare-event counts that the Poisson model fits.
- Check textbook answers: P(X = k), P(X ≤ k), P(X > k), the mean and the variance come out together, so you never have to hunt through a table.
- Decide whether the Poisson approximation of a binomial is safe: enter n and p and compare both distributions at the same point.
Example
Input
λ = 3, k = 2 (binomial comparison: n = 100, p = 0.03)
Output
P(X = 2) = 0.224042, P(X ≤ 2) = 0.423190, P(X > 2) = 0.576810, mean 3, variance 3, standard deviation 1.732051, mode 3; binomial P(X = 2) = 0.227474, difference 0.003432
For λ = 3 the value P(X = 0) = e^-3 ≈ 0.049787 is a handy sanity check on the tool.
Frequently asked questions
What real situations does the Poisson distribution describe?
Rare events counted in a fixed interval of time or space: incoming calls, radioactive decays, page views, printing defects. As long as the events are independent and the rate is stable, λ is the average count in that interval — and also the variance.
How is e^(−λ) handled without overflow?
The tool works in log space: ln P(X = k) = k·ln λ − λ − ln k!, where ln k! comes from a Lanczos approximation of ln Γ(k + 1). Even for k in the thousands nothing overflows to Infinity and back to NaN.
When may I approximate a binomial with a Poisson?
When n is large and p is small, use λ = n·p; the usual rule of thumb is n ≥ 20 and p ≤ 0.05. Smaller p and larger n shrink the error: with n = 100, p = 0.03 the gap is about 0.0034, and with n = 10000, p = 0.0003 it is an order of magnitude smaller.
Why are the mean and the variance both λ?
That is a defining property of the Poisson distribution: E[X] = Var[X] = λ. If your sample variance is clearly larger than the mean (overdispersion), the data is not really Poisson and you should reach for something wider such as the negative binomial.
Why is the mode ⌊λ⌋?
The ratio of neighbouring terms is P(X = k+1)/P(X = k) = λ/(k+1), so the probability rises while k + 1 < λ and falls afterwards; the most likely k is therefore ⌊λ⌋. When λ is an integer, ⌊λ⌋ and λ − 1 tie and both are modes — this tool reports ⌊λ⌋.
Keywords：poisson distributionprobability mass functioncumulative probabilitylambdabinomial approximation泊松分布概率质量函数累积概率期望方差二项分布近似上尾概率