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Poisson Distribution Calculator

Most Poisson pages answer one question and make you come back for the next one. This one shows the exact count, at most, fewer than, at least, more than and a band between two counts from the same rate, with the standard deviation, the most likely count and the inverse beside them: the smallest count that covers half, 95% and 99% of intervals. Every probability is built from the logarithm of the mass function and a forward recurrence, so nothing here forms a factorial that a browser cannot hold, and the upper tail is added up from its own largest term rather than taken away from 1, which is why the probability of 60 or more events at a rate of 3.5 reads 1.6963e-51 instead of zero.

Lambda 3.5P(X = 2)P(X <= 2)P(X >= 6)
lambda 3.5       mean 3.5       variance 3.5

P(X = 2)    0.1849590
P(X <= 2)   0.3208472
P(X >= 6)   0.1423864

built from the log mass function and a forward
recurrence; the upper tail is summed directly
Events

The probability of exactly that many events in one interval, which is the single term e to the minus lambda, times lambda to the count, over the count factorial. Exactly, Fewer than and At most read Count (k), More than and At least read Upper count (m), and Between reads both. Whichever is chosen, the stage prints three statements at once rather than only the one asked for.

Lambda
3.5
P(X = 2)
0.1849590
P(X <= 2)
0.3208472
P(X >= 6)
0.1423864
P(X < 2)
0.1358882
P(X > 6)
0.0652881
Standard deviation
1.8708
Most likely count
3
Median count
3
95% of intervals
at most 7
99% of intervals
at most 8

Probabilities are shown to seven decimal places and agree with exact rational arithmetic to within 1e-9. Anything below 0.000001 keeps its digits as an exponent rather than rounding away to zero, and a probability that is certain by definition is written as a plain 1 or 0. P(X = 2) = e^-3.5 * 3.5^2 / 2!

0246810121415

Each bar is the probability of exactly that many events. The shaded bar is the one count P(X = 2) covers. The most likely count is 3, at 0.2157855.

Worked examples

The cases a Poisson question usually starts from. Every figure here is computed by the same code as the answer above, so the examples and the tool can never disagree, and the rate runs up to 1,000,000 events per interval.

  • lambda 2, P(X = 0)e to the minus 2
    0.1353353
  • lambda 2, P(X <= 3)cumulative
    0.8571235
  • lambda 0degenerate
    P(X = 0) = 1, every other value 0
Accuracy. Probabilities are built from the logarithm of the mass function and a forward recurrence, so a large rate does not overflow and a far tail does not collapse to zero through cancellation. The upper tail is summed directly rather than taken as one minus the lower one, which is where the naive version loses its digits. It computes the Poisson model you specify and says nothing about whether your data are Poisson.

Common questions

How do you calculate a Poisson probability?
P(X = k) is e to the minus lambda, times lambda to the power k, divided by k factorial, where lambda is the mean number of events in one interval. At a rate of 3.5 events, P(X = 2) is 0.1849590. The page prints that line with your own numbers in it, under the answer, so a figure typed into the wrong field is easy to spot.
What is the difference between the Poisson PMF and the CDF?
The PMF is the probability of exactly one count, and the CDF is that count and everything below it added up. At a rate of 3.5, P(X = 2) is 0.1849590 while P(X <= 2) is 0.3208472: the second is the three-term sum of 0, 1 and 2, and the first is only the last of those terms. Both are on the first screen at once rather than behind a mode switch, because which one a question needs is usually clear only when you can see them side by side.
How do I find the probability of at least k events?
Put the count in the upper field and choose At least. At a rate of 3.5, P(X >= 6) is 0.1423864, and at a rate of 2, P(X >= 1) is 0.8646647. That tail is added up term by term from its own largest term rather than taken away from 1, which is what keeps a small tail readable: at a rate of 3.5 the probability of 60 or more events is 1.6963e-51, a figure one minus the cumulative probability cannot hold at all, because the cumulative side of it has already rounded to 1.
What are the mean, variance and standard deviation of a Poisson distribution?
The mean and the variance are both lambda, which is the distinguishing property of this distribution, and the standard deviation is the square root of it. At a rate of 3.5 the mean is 3.5, the variance is 3.5 and the standard deviation is 1.8708. The mean and the variance are printed on the stage and the standard deviation sits in the readout under it, because a count is easier to judge once you know how far it falls from the middle of the distribution.
How do I find the count that covers 95% of intervals?
The readout carries it: the smallest count whose cumulative probability reaches a level, which is the inverse of the CDF. At a rate of 3.5 that count is 7, because P(X <= 6) is 0.9347119, just short of 0.95, and P(X <= 7) is 0.9732611, which clears it. The median count and the 99% count sit beside it, worked out by stepping the same recurrence the sums use rather than by a normal approximation.
When is the Poisson distribution the right model?
When events arrive independently at a steady average rate and you are counting how many land in one fixed interval of time, length or area, with no upper bound on the count. A phone line taking 5 calls an hour on average gives P(X = 3) = 0.1403739 for the next hour, and P(X >= 8) = 0.1333717. If instead you have a fixed number of independent trials that each succeed with the same probability, the count is binomial rather than Poisson, and if the draws are made without replacement it is hypergeometric.
What happens when the rate is 0?
The count stops being random, and the page says so. With lambda = 0 no event ever arrives, so P(X = 0) is exactly 1 and every other count is exactly 0. Those answers come back as a plain 1 or 0 rather than a rounded decimal, and the readout marks them exact, because certainty and a probability that merely rounds to 1 at seven decimal places are different things.
How large a rate can this handle?
Up to 1,000,000 events per interval, and the figure at that ceiling is still an ordinary number: the most likely count is a million, and its probability is 0.0003989. The usual limit is the factorial. Written as e to the minus lambda times lambda to the k over k factorial, all three pieces leave the range of a browser number early: 200 factorial is already infinite, so the textbook expression returns 0 or NaN while the probability it stands for is perfectly finite. Here the mass function is computed in logarithms, which is why the probability of 200 events at a rate of 3.5 reads 2.4928e-268.
Can I enter the rate as a fraction?
Yes. A decimal like 3.5 and a fraction like 7/2 are both read, and a fraction is echoed back underneath as the lambda it was read as. A percentage is refused with the reason rather than accepted quietly, because a Poisson rate is a mean count of events and not a chance: an average of 0.3 events per interval is entered as 0.3. Grouped thousands like 1,250 are read as well.
How accurate are the probabilities?
Every figure agrees with exact rational arithmetic to within 1e-9, which is far tighter than the seven decimal places shown. The reference is not another floating-point library: it is the same probability worked out as a fraction of whole numbers, lambda to the k over k factorial times a truncated exact series for e to the minus lambda, carried out in integers of any length. At a rate of 2, P(X = 0) is e to the minus 2 and reads 0.1353353, and P(X <= 3) reads 0.8571235. That exact-rational check covers a grid of rates from 0.1 to 100, and across it the largest disagreement measured on the mass function, on the cumulative and on the tail is below 1e-11. Past 100 an exact series for e to the minus lambda is out of reach, so rates up to the 1,000,000 ceiling are checked a second way instead: the ratio of two masses, which carries no exponential at all, and it agrees to within 1e-11 as well.
Can it work out a probability between two counts?
Yes. Choose Between, and the two count fields become the ends of an inclusive band: at a rate of 3.5, P(2 <= X <= 6) is 0.7988237. The band is summed from the most likely count outward in both directions, so the largest terms go into the total first. A band entered the wrong way round is refused with the two numbers in the order they should be typed, rather than quietly returning zero.

Probabilities are built from the logarithm of the mass function and a forward recurrence, so a large rate does not overflow and a far tail does not collapse to zero through cancellation. The upper tail is summed directly rather than taken as one minus the lower one, which is where the naive version loses its digits. It computes the Poisson model you specify and says nothing about whether your data are Poisson.