Poisson Distribution Calculator

Calculate PMF, CDF, mean and variance for the Poisson distribution with rate λ.

About Poisson Distribution Calculator

The Poisson distribution models the number of events occurring in a fixed interval when events happen at a constant average rate λ and independently of each other. Both its mean and variance equal λ.

Formula

P(X=k) = (λ^k × e^(−λ)) / k!

λ: Average rate of events
k: Number of events
e: Euler's number

Worked example

With the default values below, Poisson Distribution Calculator returns P(X = k) of 0.19536681. Change any input above and every figure here is recalculated in your browser.

Inputs used

Rate (λ)minimum 0.001: 4
Number of Events (k)accepts 0–1,000,000: 3

Results

P(X = k): 0.19536681
P(X ≤ k): 0.43347012
Mean (λ): 4.0000
Variance (λ): 4.0000

Step by step

1Values used
2Formula applied
3P(X = k)= 0.19536681
4P(X ≤ k)= 0.43347012
5Mean (λ)= 4.0000
6Variance (λ)= 4.0000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

When should I use the Poisson distribution?

When counting rare events in a fixed interval (time, area, volume) where events occur independently at a constant average rate — e.g., calls per hour, defects per batch.

What is the relationship between Poisson and binomial?

The Poisson is the limit of the binomial as n→∞ and p→0 while np→λ. It approximates the binomial well when n is large and p is small.