Poisson Distribution Calculator

How it works

1. 1Enter the average event rate λ — the expected number of events per interval (e.g. 3 customer arrivals per hour).
2. 2Enter k, the exact number of events whose probability you want. The formula P(X = k) = (λᵏ · e⁻λ) / k! is applied instantly.
3. 3The bar chart plots P(X = k) for every k from 0 up to ⌈λ⌉ + 10, so you can see the full shape of the distribution.

Use cases

• Queuing theory: estimate the chance exactly 5 customers arrive in a minute when the average rate is 3 per minute.
• Reliability engineering: model the number of system failures per day and find the probability of zero failures.
• Network analysis: predict packet-arrival bursts when the mean arrival rate per millisecond is known.

Frequently asked questions

What does λ represent?

λ (lambda) is both the mean and the variance of the distribution — the average number of events expected in the chosen interval. Changing the interval length scales λ proportionally.

Does k have to be an integer?

Yes. The Poisson distribution is discrete; k must be a non-negative integer (0, 1, 2, …). The calculator truncates any decimal you enter.

Why is the mean equal to the variance?

It is a property of the Poisson process: when events occur independently at a constant average rate, both the expected value and the variance equal λ, so a single parameter characterises the distribution.