Poisson Distribution Calculator
Find exact and cumulative Poisson probabilities for any mean rate lambda and event count. Choose a single-value or range mode, scale lambda across multiple intervals, and get a full probability distribution table with worked steps.
Formula
Worked example
With lambda = 4 per hour and n = 1 interval (effective lambda = 4), and k = 2: lambda^k = 16, e^(-4) = 0.018316, k! = 2. So P(X = 2) = (16 x 0.018316) / 2 = 0.14653, or about 14.65%. The standard deviation is sqrt(4) = 2.
What the Poisson distribution models
The Poisson distribution gives the probability of a given number of independent events occurring in a fixed interval of time or space when those events happen at a known, constant average rate. Classic examples include the number of calls arriving at a call center per hour, decay events from a radioactive sample per second, server requests per minute, road accidents per week, typos per page, or customers entering a shop per day. The only shape parameter is lambda (the expected count per interval). As long as events are independent, the average rate is steady, and two events cannot occur at exactly the same instant, the Poisson model accurately captures how the actual count fluctuates around the mean.
Scaling across multiple intervals
A common need is to extend a per-interval rate over several intervals. If a machine produces defects at an average rate of 1.5 per hour and you want to model a 4-hour shift, the effective lambda for the shift is 1.5 x 4 = 6 defects. This calculator handles that scaling in the "Number of intervals" field so you never have to multiply manually. The effective lambda is what drives all probability calculations, cumulative tails, and the distribution table.
Exact, cumulative, and range probabilities
The calculator offers two modes. In exact mode it computes the point probability P(X = k), the cumulative tails P(X <= k) and P(X >= k), and the strict tails P(X < k) and P(X > k). These are useful when you want to know the chance of seeing a specific count or whether the count will fall below or above a threshold. In range mode it computes P(x1 <= X <= x2), the probability that the count falls inside a window, and the complementary outside-range probability P(X < x1 or X > x2). This is ideal for quality control tolerance bands or service-level agreement windows.
Key properties: mean, variance, and standard deviation
A defining property of the Poisson distribution is that its mean and variance are both exactly equal to lambda. This means the standard deviation is the square root of lambda. When the true rate is lambda = 9, for example, you expect 9 events but the standard deviation is 3, so values between 6 and 12 are within one standard deviation and are collectively quite probable. The distribution is right-skewed for small lambda and becomes increasingly symmetric, approaching a normal distribution, as lambda grows large. When you observe variance larger than the mean in real data, the events are clustered or correlated and a negative binomial model may fit better.
When to use Poisson vs. other distributions
Use the Poisson when you are counting events over a continuous interval with no fixed upper limit on how many can occur, and you know only the average rate. Use the binomial when there is a fixed number of independent trials each with a known success probability. The Poisson is the mathematical limit of the binomial as trials grow large and the per-trial probability shrinks, with their product held constant at lambda, which is why it is sometimes called the law of rare events. If the count data shows heavier tails than Poisson, a negative binomial distribution adds a second parameter to absorb extra variability.
Poisson point probabilities for selected lambda values
| k | lambda = 1 | lambda = 2 | lambda = 4 | lambda = 8 |
|---|---|---|---|---|
| 0 | 0.3679 | 0.1353 | 0.0183 | 0.0003 |
| 1 | 0.3679 | 0.2707 | 0.0733 | 0.0027 |
| 2 | 0.1839 | 0.2707 | 0.1465 | 0.0107 |
| 3 | 0.0613 | 0.1804 | 0.1954 | 0.0286 |
| 4 | 0.0153 | 0.0902 | 0.1954 | 0.0573 |
| 5 | 0.0031 | 0.0361 | 0.1563 | 0.0916 |
| 6 | 0.0005 | 0.0120 | 0.1042 | 0.1221 |
| 7 | 0.0001 | 0.0034 | 0.0595 | 0.1396 |
| 8 | 0.0000 | 0.0009 | 0.0298 | 0.1396 |
| 9 | 0.0000 | 0.0002 | 0.0132 | 0.1241 |
| 10 | 0.0000 | 0.0000 | 0.0053 | 0.0993 |
P(X = k) rounded to four decimal places. The mode (most probable k) is near floor(lambda).
Frequently asked questions
What is lambda in the Poisson distribution?
Lambda is the mean rate, the average number of events expected in one interval. It is the only parameter, and it sets both the center and the spread, because the variance of a Poisson distribution equals lambda as well. If events happen 4 times per hour on average, then lambda = 4 for a one-hour interval, or lambda = 8 if you measure over two hours (use the "Number of intervals" field to scale automatically).
How do I compute the probability over multiple time periods?
Enter the per-period rate in the "Mean rate per interval" field and the number of periods in the "Number of intervals" field. The calculator multiplies them to get the effective lambda. For example, if a router logs 2 errors per minute on average and you want the probability over a 5-minute window, enter lambda = 2 and intervals = 5, giving an effective lambda of 10.
What is the range probability mode?
Range mode computes P(x1 <= X <= x2), the probability that the event count falls anywhere in the window from x1 to x2 inclusive. It also reports the outside-range probability P(X < x1 or X > x2). This is useful for quality control (probability of defect count staying within tolerance) or forecasting (probability of demand falling within a target band).
Why must k be a whole number?
The Poisson distribution counts discrete events, so k must be a non-negative integer. You cannot have 2.5 phone calls or 3.7 server requests. The rate lambda, however, can be any non-negative real number because it is an average computed across many intervals, not a single observed count.
What is the standard deviation of a Poisson distribution?
Because the variance of a Poisson equals its mean lambda, the standard deviation is simply sqrt(lambda). When lambda = 9, the standard deviation is 3. This property means the coefficient of variation (std dev / mean) is 1 / sqrt(lambda), so relative variability shrinks as the rate grows large.
When should I use a Poisson distribution instead of a binomial?
Use the Poisson when you are counting events over a continuous interval with no fixed upper limit on the count, and you know only the average rate. Use the binomial when there is a fixed number of independent trials each with a success probability. In practice the Poisson is the limiting case of the binomial as trials grow large and the per-trial probability shrinks while their product (the mean) stays fixed.