Statistics and research
Poisson probability calculator
Poisson probabilities P(X = k), P(X ≤ k) and P(X ≥ k) for a mean rate λ, with the distribution table, accurate in the far tails.
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Results
P(X = 5)0.132169P(X ≤ 5)0.857614P(X ≥ 5)0.274555
- P(X = 5)
- 0.132168599786
- P(X < 5)
- 0.72544495331
- P(X ≤ 5)
- 0.857613553096
- P(X > 5)
- 0.142386446904
- P(X ≥ 5)
- 0.27455504669
- Mean and variance λ
- 3.5
- Standard deviation √λ
- 1.870828693
| x | P(X = x) | P(X ≤ x) | P(X ≥ x) |
|---|---|---|---|
| 0 | 0.030197383 | 0.030197383 | 1 |
| 1 | 0.10569084 | 0.13588823 | 0.96980262 |
| 2 | 0.18495897 | 0.3208472 | 0.86411177 |
| 3 | 0.21578547 | 0.53663267 | 0.6791528 |
| 4 | 0.18881229 | 0.72544495 | 0.46336733 |
| 5 | 0.1321686 | 0.85761355 | 0.27455505 |
| 6 | 0.07709835 | 0.9347119 | 0.14238645 |
| 7 | 0.038549175 | 0.97326108 | 0.065288097 |
| 8 | 0.016865264 | 0.99012634 | 0.026738922 |
| 9 | 0.0065587138 | 0.99668506 | 0.0098736581 |
| 10 | 0.0022955498 | 0.99898061 | 0.0033149443 |
| 11 | 0.00073040222 | 0.99971101 | 0.0010193944 |
| 12 | 0.00021303398 | 0.99992404 | 0.00028899222 |
| 13 | 5.7355302e-5 | 0.9999814 | 7.595824e-5 |
| 14 | 1.4338826e-5 | 0.99999574 | 1.8602937e-5 |
| 15 | 3.345726e-6 | 0.99999908 | 4.2641118e-6 |
| 16 | 7.3187756e-7 | 0.99999981 | 9.1838586e-7 |
| 17 | 1.5068067e-7 | 0.99999996 | 1.8650831e-7 |
| 18 | 2.929902e-8 | 0.99999999 | 3.5827634e-8 |
Cumulative probabilities come from the regularized incomplete gamma function, P(X ≤ k) = Q(k + 1, λ).
Method and limits
How to use it
Enter the mean number of events λ in the interval of interest (rate × length of interval) and a count k. You get exact and cumulative probabilities and the distribution table.
Formulas
- P(X = k) = λk e−λ / k!, computed in Loader’s saddle-point form.
- P(X ≤ k) = Q(k + 1, λ), the regularized upper incomplete gamma function; P(X > k) = P(k + 1, λ), computed directly so tiny tails stay accurate.
- Mean and variance λ.
Assumptions and limits
- Events occur independently at a constant average rate; two events cannot happen at exactly the same instant.
- If the variance of real counts is well above the mean (overdispersion), the Poisson model understates the spread.
- λ up to 10 million.
Worked example
A help desk receives 3.5 calls per hour on average. The chance of exactly 5 calls in an hour is 0.1322, of at most 5 is 0.8576, and of 5 or more is 0.2746.
References
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