Poisson Distribution Calculator
Calculate Poisson probabilities for events occurring in a fixed interval.
Probability
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Enter λ and k
P(X = k)
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P(X ≤ k)
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Mean (μ)
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Variance (σ²)
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Poisson Distribution
The Poisson distribution models the probability of a given number of events occurring in a fixed interval, given a known average rate.
Probability Mass Function
P(X = k) = (λ^k × e^(-λ)) / k!
Properties
- Mean: E(X) = λ
- Variance: Var(X) = λ
- Support: k = 0, 1, 2, ...
Applications
| Application | Example |
|---|---|
| Call Centers | Calls per hour |
| Quality Control | Defects per batch |
| Traffic | Cars at intersection per minute |
| Biology | Mutations per DNA segment |
Conditions for Poisson
- Events occur independently
- Rate is constant
- Two events cannot occur at exactly the same instant
- Events are rare relative to the interval
Frequently Asked Questions
When to use Poisson vs Binomial?
Use Poisson when counting events over time/space with a known average rate. Use Binomial when you have a fixed number of trials with two outcomes.
What is the relationship between Poisson and Exponential?
If events follow a Poisson process with rate λ, the time between events follows an Exponential distribution with parameter λ.