04 // Interactive Formula Compendium // STATISTICSP(X = k) = (λ^k · e^(−λ)) / k!
Poisson Rare-Event Distribution Formula
The Poisson PMF P(X=k) = λ^k·e^(−λ)/k! calculates the probability of k independent events in an interval with mean arrival rate λ, uniquely featuring equal mean and variance (μ = σ² = λ).
Formula Reference
Main Formula
P(X = k) = (λ^k · e^(−λ)) / k!
Rearranged Forms
E[X] = Var(X) = λ
T_interarrival ~ Exponential(λ)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| P | Event Count Probability P(X = k) | probability (0–1) |
| lam | Mean Rate λ | events/interval |
| k | Observed Count k | events |
Inputs & Parameters
[events/interval]
[events]
Calculated Result (Event Count Probability P(X = k))
P =0.156294probability (0–1)
Formula Notes & Background
- Derived as the limit of Binomial(n, λ/n) as n → ∞: C(n,k)(λ/n)^k(1 − λ/n)^(n−k) → (λ^k/k!)·e^(−λ).
- Waiting times between consecutive Poisson arrivals follow an Exponential distribution f(t) = λe^(−λt).
Step-by-Step Examples
#1: An API gateway averages λ = 4.0 error events per minute. Find the probability of observing exactly k = 5 errors in the next minute.
λ = 4.0k = 5
P(X = 5) = (4⁵ × e^(−4)) / 120 = (1024 × 0.018316) / 120
→ P(X = 5) = 0.1563 (15.63%)
#2: Probability of zero errors (k = 0) during a 1-minute interval with λ = 4.0.
λ = 4.0k = 0
P(X = 0) = e^(−4.0)
→ P(X = 0) = 0.0183 (1.83%)
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Frequently Asked Questions
How can I check empirically if count data is Poisson-distributed?+
Check the dispersion index Var(X)/E[X]: a true Poisson process has ratio ≈ 1.0 (overdispersed > 1 suggests a Negative Binomial model).
Does the sum of two independent Poisson variables stay Poisson?+
Yes: if X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are independent, X + Y ~ Poisson(λ₁ + λ₂).
Why use log-factorials in numerical evaluation?+
Direct k! overflows IEEE-754 64-bit floats at k = 171, whereas exp(k·ln(λ) − λ − ln(k!)) remains exact for large k.