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Poisson Distribution Calculator

Enter the average number of events λ in an interval to find the probability of exactly k events, at most k, at least k or a range. Results come with the mean, standard deviation, a probability table and a highlighted chart.

Average rate (λ)

Expected number of events in the interval

Number of events (k)

From (a)

To (b)

Enter your values to see the result.

How to use the Poisson Distribution Calculator

Enter λ, the average number of events per interval (e.g. 4 calls per hour).
Choose the probability: exactly, at most, at least or between.
Enter k (or a and b).
Press Calculate to get the probability and summary statistics.
Read the chart and table, or download the table as CSV.

Poisson distribution formulas

Quantity	Formula
Probability of exactly k events	P(X = k) = e^(−λ) λᵏ / k!
At most k (CDF)	P(X ≤ k) = Σ_i=0^k e^(−λ) λⁱ / i!
At least k	P(X ≥ k) = 1 − P(X ≤ k − 1)
Mean and variance	E[X] = Var(X) = λ
No events	P(X = 0) = e^(−λ)
Recursion	P(X = k + 1) = P(X = k) · λ / (k + 1)

Probabilities for common rates

k	λ = 0.5	λ = 1	λ = 2	λ = 4
0	0.6065	0.3679	0.1353	0.0183
1	0.3033	0.3679	0.2707	0.0733
2	0.0758	0.1839	0.2707	0.1465
3	0.0126	0.0613	0.1804	0.1954
4	0.0016	0.0153	0.0902	0.1954
5	0.0002	0.0031	0.0361	0.1563
6	0.0000	0.0005	0.0120	0.1042
7	0.0000	0.0001	0.0034	0.0595

Conditions for a Poisson model

Events occur one at a time and independently of each other.
The average rate λ is constant over the interval (scale it for other intervals: 4 per hour = 2 per half hour).
Two events cannot happen at exactly the same instant.
If the variance of your data is much larger than its mean, the data are over-dispersed and a Poisson model may underestimate extreme counts.

Frequently asked questions

What is the Poisson distribution?

It models the number of events in a fixed interval of time or space when events happen independently at a constant average rate λ, for example calls to a help desk per hour, typos per page or goals in a match.

What is the Poisson formula?

P(X = k) = e^(−λ) λᵏ / k!. With λ = 4, the probability of exactly 6 events is e^(−4) × 4⁶ / 720 ≈ 0.104196.

What are the mean and variance of a Poisson distribution?

Both equal λ, so the standard deviation is √λ. If a shop gets 9 customers per hour on average, the count per hour has mean 9 and standard deviation 3.

How do I calculate P(X ≥ k)?

Use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1). For example with λ = 10, P(X ≥ 15) = 1 − P(X ≤ 14) ≈ 0.083458. This calculator sums the smaller tail directly, so even very small probabilities stay accurate.

What is the probability of zero events?

P(X = 0) = e^(−λ). With λ = 0.5 typos per page, the chance a page has no typos is e^(−0.5) ≈ 60.65%.

When should I use Poisson instead of binomial?

Use Poisson when you know an average rate but there is no fixed number of trials, or when n is large and p is small. A binomial with n trials and small p is approximately Poisson with λ = np.

Can λ be a decimal?

Yes. λ is an average and can be any positive number, such as 2.7 goals per match. The number of events k must be a whole number.

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