Poisson Distribution Calculator
Poisson Distribution Calculator
Find the exact chance of a count of independent events that happen at a steady average rate: exactly, at most, at least or between. Work from a rate and a window, get the waiting time to the next event, an exact interval for a rate from a count, or compare with the binomial.
Distribution
P(X = j) = e^−λ × λ^j ÷ j!, with mean and variance both equal to λ. Cumulative answers add terms outward from the most likely count, each found from logarithms, so a mean of a million works without overflow, and a tail is never found by subtracting from 1. The wait for the next event at rate r is exponential: P(wait ≤ t) = 1 − e^−rt, median ln 2 ÷ r, mean 1 ÷ r. The exact interval for a count k uses chi square quantiles: lower = χ²(α/2, 2k) ÷ 2 and upper = χ²(1 − α/2, 2k + 2) ÷ 2. Events are assumed independent, one at a time, at a constant average rate. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Choose what you want in the Calculate box. Probabilities from a mean is the classic case: enter the average number of events λ, pick exactly, at most, fewer than, at least, more than or between, and enter x. Rate and window multiplies a rate by a window length to give λ, and the unit is only a label. The other modes give the wait for the next event, a count or a mean for a target chance, an exact interval from a count, a binomial comparison and a full table.
The input people most often get wrong is the window. The mean must belong to the same stretch of time, length or area as the count you ask about, so a rate of 12 per hour over 30 minutes has a mean of 6, not 360, and the rate and window boxes only agree if both use the same unit. The second trap is the condition. At least 5 includes 5 and more than 5 does not; at a mean of 3 they are 18.47% and 8.39%.
What the calculation is doing
A count X is Poisson when events happen one at a time, independently of one another, at a steady average rate. Then P(X = x) = e^−λ × λ^x ÷ x!, where λ is the mean number of events in the window. The variance is also λ, so the standard deviation is √λ, and that equality is the first thing to check against real data. The chance of no events is e^−λ, and the chance of at least one is 1 − e^−λ.
Done naively, this fails at large means: e^−λ underflows to zero past λ of about 745, and λ^x or x! overflows sooner. The calculator works from logarithms with the saddle point method R uses, and adds each tail outward from the most likely count. At a mean of 1,000,000 the chance of 1,003,000 or more comes out as 0.136% instead of an error, and no tail is found by subtracting from 1.
The other modes use the same model. A rate r over a window t gives λ = r × t. The wait for the next event is exponential: P(wait ≤ t) = 1 − e^−rt, with mean 1 ÷ r and median ln 2 ÷ r. The exact interval for a count k uses chi square quantiles, lower = χ²(α/2, 2k) ÷ 2 and upper = χ²(1 − α/2, 2k + 2) ÷ 2. Each limit is the mean at which seeing k or more, or k or fewer, has probability α/2.
Three conditions, and how real data break them
Each condition fails in a recognizable way. Independence fails when events cluster: earthquakes bring aftershocks, and the USGS declusters its catalogs because the Poisson model needs independent events. A steady rate fails when demand has a rhythm, such as calls that peak at ten in the morning. One at a time fails when arrivals come in groups, such as a busload of passengers. In that case count the groups instead.
The variance gives the fastest check. The ratio of variance to mean, the index of dispersion, is 1 for a Poisson. Above 1 is overdispersion, more very quiet and very busy intervals than a Poisson allows, which is what clustering looks like. Below 1 is more regular than random. Bortkiewicz's Prussian cavalry data fit well: 122 deaths in 200 corps years, a mean of 0.610 and a variance of 0.611. As a formal test, Σ(x − mean)² ÷ mean is compared with chi square on n − 1 degrees of freedom, and Hoel found that reasonable once the mean is above 5. When the ratio is well above 1, treat Poisson answers as a best case and consider a negative binomial.
Rare events and the binomial
The Poisson is the limit of the binomial as n grows and p shrinks with np held fixed. Penn State's rule of thumb is that it works well if n ≥ 20 and p ≤ 0.05, or n ≥ 100 and p ≤ 0.10. For 100 bulbs at 5% defective, at most 3 defective is 25.78% exactly and 26.5% by Poisson, matching the 0.258 and 0.265 that Penn State prints.
Le Cam's theorem guarantees that the sum of the absolute gaps between the two distributions is at most 2np². With n = 1,000 and p = 0.002 the bound is 0.008 and the actual total is 0.000903. It is a guarantee, not an estimate, and about 9 times loose here.
The wait for the next event, and why a return period is not a schedule
If the count in any window of length t is Poisson with mean rt, the gaps between events are exponential with mean 1 ÷ r. At 4 per hour the mean gap is 15 minutes, the median is 10.4 minutes, and the chance the next event arrives within 30 minutes is 86.47%. The median sits below the mean because a few long gaps pull the mean up. The wait is memoryless: 20 minutes without an event does not make one due.
A return period is an average gap, not a timetable. The USGS defines a 100 year flood as one with a 1 percent chance in any year, and each year begins with the same chance. That is a rate of 0.01 per year, so over 100 years the mean count is 1: the chance of none is 36.79% and of two or more is 26.42%. Over 30 years the chance of at least one is 25.92% by Poisson, or 26.03% counting each year as a 1% trial.
What a count of zero still tells you
Counting no events does not mean the rate is zero. For k = 0 the exact two sided limit runs from 0 to −ln(α/2), which is 3.6889 at 95%, and the one sided limit is −ln(0.05) = 2.9957, the rule of three: if nothing happened in n trials, 0 to 3 ÷ n is a 95% interval for the rate. No defects in 120 welds still allows a defect rate near 2.5%.
For a positive count the interval is uneven. Fourteen events in 400 person years is a rate of 0.035, with exact 95% limits of 0.019135 to 0.058724, the figures StatsDirect prints. The shortcut k ± z√k gives 6.666 to 21.33 for the count against the exact 7.6539 to 23.49, too narrow on the upper side. The exact interval is conservative: its coverage is never below the stated level.
Reference tables: none, at least one, and exact limits
The first table gives the chance of none and of at least one for common means. The second gives exact 95% limits after 0 to 10 events, matching the McMaster course table to four decimals.
| Mean, λ | None, e^−λ | At least one, 1 − e^−λ |
|---|---|---|
| 0.1 | 90.48% | 9.52% |
| 0.5 | 60.65% | 39.35% |
| 1 | 36.79% | 63.21% |
| 2 | 13.53% | 86.47% |
| 3 | 4.98% | 95.02% |
| 5 | 0.674% | 99.33% |
| 7 | 0.0912% | 99.91% |
| 10 | 0.00454% | 99.99546% |
| 15 | 0.0000306% | 99.9999694% |
| 20 | 2.06 × 10^−7% | 99.999999794% |
| Count, k | Lower limit | Upper limit | One sided upper |
|---|---|---|---|
| 0 | 0 | 3.6889 | 2.9957 |
| 1 | 0.025318 | 5.5716 | 4.7439 |
| 2 | 0.24221 | 7.2247 | 6.2958 |
| 3 | 0.61867 | 8.7673 | 7.7537 |
| 4 | 1.0899 | 10.242 | 9.1535 |
| 5 | 1.6235 | 11.668 | 10.513 |
| 6 | 2.2019 | 13.059 | 11.842 |
| 7 | 2.8144 | 14.423 | 13.148 |
| 8 | 3.4538 | 15.763 | 14.435 |
| 9 | 4.1154 | 17.085 | 15.705 |
| 10 | 4.7954 | 18.39 | 16.962 |
A worked example you can check on paper
The calculator opens on a mean of 3 and the chance of at least 5 events. First e^−3 = 0.0497871. Each next term is the last one times 3 ÷ (j + 1), giving P(0) to P(4) = 0.0497871, 0.149361, 0.224042, 0.224042, 0.168031. They sum to 0.815263, so P(X ≥ 5) = 1 − 0.815263 = 0.184737, or 18.47%. The calculator adds P(5) + P(6) + P(7) and so on instead, because subtraction is only safe when the answer is not small.
The mean and variance are both 3, so the standard deviation is √3 = 1.7321, and 5 is 1.15 standard deviations above the mean. The most likely counts are 2 and 3, each 22.4%, because λ is a whole number.
Mistakes that catch people out
Mixing units. A rate of 12 per hour and a window of 30 minutes are not the same units. Convert first: 0.5 hours gives a mean of 6, and typing 30 gives 360.
Confusing at least with more than. P(X ≥ 5) includes 5 and P(X > 5) starts at 6. At a mean of 3 they are 18.47% and 8.39%.
Reading a return period as a schedule. A 1 in 100 year event has a 25.92% chance of showing up within any 30 years, and can happen two years running.
Ignoring clusters. If the variance of your counts is much larger than their mean, events are not independent and the Poisson answers are too optimistic about ordinary periods. Check the ratio first.
Treating zero as none. Zero events in an exposure bounds the rate from above but does not pin it to zero: the exact 95% upper limit is 3.6889 events for the whole exposure.
Popular calculations
Each opens the calculator with those numbers filled in, and they can be changed once it is open.
Probabilities from a mean
Rates and windows
Waiting for the next event
Finding a count or a mean for a target chance
Exact intervals from a count
Poisson against the binomial
Distribution tables
How this was checked
A separate script compared the engine with independent routes. The terms add to 1 within 4.2 × 10^−15 for 20 means from 0.001 to 1,000,000. Cumulative and tail sums match 70 digit decimal arithmetic at 170 points, to about 10^−13 for answers above e^−1000, and match scipy to 2 × 10^−13 up to a mean of 20,000. The count for a target chance matched scipy on 121 cases, and 300 random round trips found no violation. P(no event in t) equalled P(X = 0) in 500 random cases.
The exact limits reproduce the McMaster table for counts 0 to 6 (7 of 7 rows), the GraphPad limits for a count of 25 at three levels, and the StatsDirect interval for 14 events in 400 person years. The chi square quantiles match the NIST table in 20 of 20 entries and scipy to 1.5 × 10^−15. The Le Cam and Chen bounds held in all 156 pairs of n and p. OpenStax Examples 4.29 to 4.32, the Penn State bulb example and all five Bortkiewicz expected counts reproduce. A sweep of 2,916 answers found no NaN, no Infinity and no probability printed as 100% when it was not exactly 1. Every popular address was opened and checked for a distinct title, the right answer and clean exports.
Common questions
What is the Poisson distribution used for?
It models the number of events in a fixed window when events arrive one at a time, independently, at a steady average rate: calls per hour, defects per metre, faults per year. Given the mean it gives the chance of every count. With a mean of 3, exactly 5 events has a chance of 10.08%.
How do you find the probability of at least one event?
Work out the chance of none, e^−λ, and take it from 1. With a mean of 3 the chance of none is 4.98%, so at least one is 95.02%. For at least 5 or more, add the tail terms as the calculator does, because subtraction loses digits when the answer is tiny.
What is the difference between the Poisson and the binomial distribution?
The binomial counts successes in a fixed number of trials, so it never exceeds n, and its variance is np(1 − p). The Poisson has no upper limit and its variance equals its mean. With many trials and a small p they nearly agree with λ = np: 100 bulbs at 5% defective, at most 3, is 25.78% exactly and 26.5% by Poisson.
How do I know whether my counts are Poisson?
Compare the variance of past counts with their mean; for a Poisson the ratio is 1. Well above 1 signals clustering, and below 1 signals counts more regular than random. Bortkiewicz's horse kick data give a mean of 0.610 and a variance of 0.611.
How long until the next event?
If events follow a Poisson process at rate r, the wait is exponential with mean 1 ÷ r and median ln 2 ÷ r. At 4 per hour the mean wait is 15 minutes, the median is 10.4 minutes, and the chance the next event arrives within 30 minutes is 86.47%. The wait is memoryless, so time already spent does not shorten what remains.
What does a confidence interval for a count of zero mean?
Zero events does not prove the rate is zero. The exact 95% interval for the mean count runs from 0 to 3.6889, and the one sided 95% upper limit is 2.9957, the rule of three. If nothing went wrong in 120 inspections, the failure rate per inspection could still be about 2.5%.
Related Calculators and Reference
- Binomial Distribution Calculator: the exact answer for a fixed number of trials, which the Poisson only approximates.
- Confidence Interval Calculator: intervals for a mean or a proportion once counts are large enough for the normal shortcut.
- Standard Deviation and Variance Calculator: work out the variance of past counts to test the variance equals mean rule.
- Chi Square Calculator: test whether observed counts match the frequencies a Poisson predicts.
- Z Score Calculator: the normal curve that approximates a Poisson at large means.
- NIST/SEMATECH Handbook of Statistical Methods: Poisson Distribution: the formulas for the probability, cumulative distribution, mean, standard deviation and mode.