Poisson Distribution Calculator

Use our free Poisson distribution calculator to find P(X=k), P(X≤k), and P(X≥k) for a given rate λ with mean and variance.

Math

Probability Type

Rate (λ)

Occurrences (k)

Formula

The Poisson distribution models the number of events occurring in a fixed interval given a known average rate λ.

P(X=k) = e^(-λ) × λ^k / k!

Percentage: The percentage value you want to apply

Number: The original number or value

Result: The calculated result

How It Works

1Select exact, at-most, or at-least.
2Enter λ (rate) and k (occurrences).
3Click Calculate.

Examples

λ=3, k=2

1.P(X=2)=e⁻³×3²/2!≈0.2240
2.Mean=3, Var=3, SD≈1.732

Result: 0.224042

λ=3, k≤2

1.Sum P(0)+P(1)+P(2)

Result: 0.423190

Frequently Asked Questions

Models count of events in a fixed interval at average rate λ.

P(X=k) = e^(-λ) × λ^k / k!.

Both equal λ.

Events independent, constant rate, no overlap.

Poisson for rare events with large n, small p. Approximates binomial when n large, p small.

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CalculateMe Team

Last updated: 2026-07-14

Poisson Distribution Calculator

Formula

How It Works

Examples

λ=3, k=2

λ=3, k≤2

Frequently Asked Questions

What is Poisson distribution?

Formula?

Mean and variance?

Requirements?

Poisson vs binomial?

Related Calculators

Poisson Distribution Calculator

Formula

How It Works

Examples

λ=3, k=2

λ=3, k≤2

Frequently Asked Questions

What is Poisson distribution?

Formula?

Mean and variance?

Requirements?

Poisson vs binomial?

Related Calculators