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Poisson Distribution Calculator

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

Inputs

P(X = k)

0.19536681

P(X ≤ k)

0.43347012

Mean (λ)

4.0000

Variance (λ)

4.0000

Step by step

  1. Values used

    Rate (λ) = 4; Number of Events (k) = 3

  2. Formula applied

    P(X=k) = (λ^k × e^(−λ)) / k!

  3. P(X = k)

    = 0.19536681

  4. P(X ≤ k)

    = 0.43347012

  5. Mean (λ)

    = 4.0000

  6. Variance (λ)

    = 4.0000

How it works

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

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.

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