Poisson Distribution Calculator: PMF, CDF, Table, and Inverse
Compute Poisson probabilities for any comparison, read the complete k table around lambda, get the bracketed inverse with both attained probabilities, and see the normal approximation error.
At a glance
- Computes
- Exact Poisson probabilities, moments, and the bracketed inverse
- You supply
- A mean count lambda, or a rate and an interval, plus a count
- Use when
- Counting events in a fixed interval when there is no fixed number of trials
- Not for
- A fixed number of independent trials counted as successes Binomial Distribution Calculator
P(X ≤ 3)
0.757576
75.758%
Density by events; 10 bins of width 1 (fd rule); Poisson distribution with lambda = 2.5; P(X ≤ 3)
0.757576
| Detail | Value |
|---|---|
| Lambda | 2.50000 |
| Mean | 2.50000 |
| Variance | 2.50000 |
| Standard deviation | 1.58114 |
| Mode | 2 |
| Skewness | 0.632456 |
| Excess kurtosis | 0.400000 |
| Index of dispersion | 1 |
Exact result and approximation
| Method | Value | Absolute error | Relative error |
|---|---|---|---|
| Exact | 0.757576 | reference | reference |
| Normal with continuity correction | 0.7364553716 | 0.02112 | 2.788% |
| Normal without correction | 0.6240851830 | 0.1335 | 17.62% |
Lambda thresholds below are named conventions. Their observed error is the result that matters.
- lambda ≥ 10
- lambda = 2.50000; not met
- lambda ≥ 5
- lambda = 2.50000; not met
Bracketed inverse
Requested probabilities within about 1e-11 of a value the distribution actually attains are treated as that value, because the cumulative probability itself is only accurate to about 1e-13.
| Detail | Value |
|---|
Count table
Table cells show 6 significant figures. Mass below the window: 0. Mass above the window: 0.000277. The Poisson support has no upper bound, so a window is always a window.
| k | P(X = k) | P(X ≤ k) | P(X ≥ k) | In region |
|---|---|---|---|---|
| 0 | 0.0820850 | 0.0820850 | 1 | yes |
| 1 | 0.205212 | 0.287297 | 0.917915 | yes |
| 2 | 0.256516 | 0.543813 | 0.712703 | yes |
| 3 | 0.213763 | 0.757576 | 0.456187 | yes |
| 4 | 0.133602 | 0.891178 | 0.242424 | |
| 5 | 0.0668009 | 0.957979 | 0.108822 | |
| 6 | 0.0278337 | 0.985813 | 0.0420210 | |
| 7 | 0.00994062 | 0.995753 | 0.0141873 | |
| 8 | 0.00310644 | 0.998860 | 0.00424670 | |
| 9 | 0.000862901 | 0.999723 | 0.00114025 |
Equivalent calls
R ppois(3, 2.5) Python scipy.stats.poisson.cdf(3, 2.5) Excel =POISSON.DIST(3, 2.5, TRUE) TI-84 poissoncdf(2.5, 3)
Lambda is a rate over a matching interval
This page counts events in a fixed interval of time, area, volume or exposure when there is no fixed number of trials. Enter lambda directly or make the multiplication visible with rate and exposure. Use matching units because the calculator does not perform a unit conversion.
When the number of trials is fixed and each trial has one success probability, use the binomial distribution page. That page also owns the Poisson approximation to the binomial. A variance above the mean indicates overdispersion, where its negative binomial mode is often more appropriate.
P(X = k) = exp(-lambda) lambda^k / k!
How?
How this is calculated
Masses are evaluated in log space. Cumulative and survival probabilities use direct checked incomplete-gamma tails, and outside ranges add the two tails. To test observed counts against expected counts, use the chi-square test. Interval estimation belongs on the confidence intervals page.
Formula: P(X = k) = exp(-lambda) lambda^k / k!
Why the approximation rules are conventions
References use both lambda 5 and lambda 10 as working thresholds. This tool evaluates both and prints the normal approximation's actual error against the exact result. A threshold is guidance, not a guarantee.
Sources
- NIST SEMATECH e-Handbook, Poisson distribution (1.3.6.6.19). NIST/SEMATECH. Retrieved .
- R documentation, The Poisson Distribution. R Core Team. Retrieved .