Poisson Distribution Calculator
Poisson Distribution Calculator
Calculate exact, cumulative, tail, and interval probabilities for Poisson count data. Results update instantly as values change.
Inputs
The slider covers λ = 0–100. The number field supports values up to 10,000.
λ = rate × exposure = 2 × 2 = 4
Percentile / quantile
8
smallest x with F(x) ≥ 95%
Live result
Probability
0.146525
P(X = 2)
Percentage
14.6525%
same probability as a percentage
Complement
0.853475
1 − selected probability
Poisson probability massBlue marks the selected event
Selected vs. complementlive
Mean
4
Variance
4
Std. dev.
2
Skewness
0.5
Excess kurtosis
0.25
Use a Poisson model when counts represent events in a fixed interval or exposure, events occur independently, and the average event rate is reasonably constant. The model also assumes the probability of two or more events in an extremely small interval is negligible.
How this is calculated
The Poisson probability mass function is P(X = k) = e−λ λk / k!. Cumulative and tail probabilities are built from this same distribution.
P(X = 2) = e^(−4) × 4^2 / 2! = 0.1465251111
λ = rate × exposure
For a Poisson variable, mean = variance = λ and standard deviation = √λ.
This Poisson Distribution Calculator helps researchers, students, and analysts estimate the probability of a specified number of events when occurrences follow a Poisson process. Enter the expected mean count λ directly, or calculate λ from an event rate and exposure. The tool supports exact probabilities, cumulative probabilities, upper and lower tails, inclusive count ranges, and Poisson percentiles. It also displays the mean, variance, standard deviation, skewness, excess kurtosis, a live probability chart, and the complement of the selected event. Because every result is calculated in the browser with the standard Poisson probability formula, no dataset upload or external calculation service is required.
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