Poisson Distribution
Print PageA Poisson Distribution Calculator (also known as a Poisson Probability Calculator, Rare Event Count Utility, Poisson CDF Analyzer, or Event Rate Parameter [λ] Calculator) computes exact single-point probabilities (P(X = k) = [ λk · e-λ ] ÷ k!), lower cumulative probabilities (P(X ≤ k)), upper tail probabilities (P(X ≥ k)), expected mean (μ = λ), variance (σ2 = λ), and standard deviation (σ = √λ) for discrete independent events occurring at a constant average rate λ in a fixed interval of time or space.
In web server capacity planning, call center staffing, highway traffic toll design, and radioactive decay tracking, the Poisson distribution models rare independent event arrivals where occurrences cannot happen simultaneously and mean equals variance.
Our free online Poisson Distribution Calculator provides instant calculations across all discrete rate parameters:
- Probability Mass Function (PMF for k = 0, 1, 2…):
P(X = k) = [ λk · e-λ ] ÷ k!. - Lower Cumulative Distribution Function (CDF / P[X ≤ k]):
P(X ≤ k) = ∑j=0k [ λj · e-λ ] ÷ j!. - Upper Tail Event Probability (P[X ≥ k]):
P(X ≥ k) = 1 - P(X ≤ k - 1). - Mean-Variance Equality Property:
μ = λandσ2 = λ. - Standard Deviation (σ):
σ = √λ. - Binomial Approximation Limit (n ≥ 20, p ≤ 0.05):
Binomial(n, p) ≈ Poisson( λ = n · p ).
Master Poisson Reference Table (λ = 4.0 Server Requests per Second)
The table below displays single-point probabilities P(X = k), lower cumulative probabilities P(X ≤ k), upper tail probabilities P(X ≥ k), and operational interpretations for an e-commerce web server receiving an average rate of λ = 4.0 requests per second:
| Number of Events (k Requests) | PMF Factorial Formula | Exact P(X = k) | Lower Cumulative P(X ≤ k) | Upper Tail P(X ≥ k) | Server Capacity Interpretation |
|---|---|---|---|---|---|
| k = 0 Requests (Idle Second) | [ 4.00 · e-4.0 ] ÷ 0! | 1.8316% (0.01832) | 1.8316% | 100.0000% | Idle Idle Second Probability |
| k = 2 Requests | [ 4.02 · e-4.0 ] ÷ 2! | 14.6526% (0.14653) | 23.8103% (23.81%) | 90.8422% | 23.8% Chance of ≤2 Requests |
| k = 4 Requests (μ = λ Mean Peak) | [ 4.04 · e-4.0 ] ÷ 4! | 19.5367% (0.19537) | 62.8837% (62.88%) | 56.6530% | Peak Single-Count Probability |
| k = 5 Requests (≥5 Spike Threshold) | [ 4.05 · e-4.0 ] ÷ 5! | 15.6293% (0.15629) | 78.5130% | 37.1163% (37.12%) | 37.1% High Load Spike Probability |
| k = 8 Requests (Heavy Traffic Spike) | [ 4.08 · e-4.0 ] ÷ 8! | 2.9770% (0.02977) | 97.8637% (97.86%) | 5.1133% | Heavy Spike Traffic Event (2.98%) |
Step-by-Step Server Request Probability Calculation (λ = 4.0)
To evaluate web server traffic for a site receiving an average rate of λ = 4.0 requests per second:
Step 1 (Calculate Probability of Zero Requests P[X = 0]): P(X = 0) = (4.0^0 · e^-4.0) ÷ 0! = 1 · 0.0183156 ÷ 1 = 0.018316 ≈ 1.83%
Step 2 (Calculate Probability of Exactly 4 Requests P[X = 4]): P(X = 4) = (4.0^4 · e^-4.0) ÷ 4! = (256 · 0.0183156) ÷ 24 = 4.6888 ÷ 24 = 0.195367 ≈ 19.54%
Step 3 (Calculate Cumulative Probability of ≤2 Requests P[X ≤ 2]): P(X ≤ 2) = P(0) + P(1) + P(2) = 0.01832 + 0.07326 + 0.14653 = 0.238103 ≈ 23.81%
Step 4 (Calculate Upper Spike Probability P[X ≥ 5]): P(X ≥ 5) = 1 - P(X ≤ 4) = 1 - 0.628837 = 0.371163 ≈ 37.12%
Step 5 (Verify Mean and Standard Deviation): μ = 4.0, σ^2 = 4.0 &implies; σ = √4.0 = 2.00 requests per second
Thus, the server has a 19.54% chance of receiving exactly 4 requests in any second and a 37.12% chance of experiencing a high-traffic surge of 5+ requests.
Probability Models Comparison: Poisson vs. Binomial vs. Exponential vs. Normal
Below is a comparative reference chart detailing when to use the Poisson distribution versus related models:
| Probability Model | Target Variable Type | Core Constraint / Property | Primary Practical Application |
|---|---|---|---|
| Poisson Distribution | Discrete Count k of rare events | Fixed rate λ (μ = σ2 = λ) | Website hits, customer arrivals, call centers. |
| Binomial Distribution | Discrete Count k of successes in n trials | Fixed trial count n & probability p | Quality control defects per batch size n. |
| Exponential Distribution | Continuous Time x BETWEEN Poisson events | Memoryless continuous waiting time | Component lifespan & service inter-arrival time. |
| Normal Distribution | Continuous symmetric variable X | Symmetric bell curve centered at μ | Poisson limit approximation for large λ (>20). |
History & Mathematics: 1837 Siméon Poisson to 1898 Bortkiewicz
1837 Siméon Denis Poisson & Law of Small Numbers
In 1837, French mathematician Siméon Denis Poisson published the distribution in Recherches sur la probabilité des jugements as the limiting mathematical case of the binomial distribution for rare events.
1898 Ladislaus Bortkiewicz & Prussian Army Horse Kicks
In 1898, Russian-German economist Ladislaus Bortkiewicz published Das Gesetz der kleinen Zahlen (The Law of Small Numbers), proving the practical real-world accuracy of Poisson modeling by tracking the annual deaths of Prussian soldiers caused by accidental horse kicks.
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Frequently Asked Questions (FAQ)
What is the Poisson Distribution formula?
The single-point probability formula is P(X = k) = [ λk · e-λ ] ÷ k!.
Why is the variance equal to the mean in a Poisson distribution?
In a Poisson process, events occur independently at a constant rate λ, mathematically forcing the mean and variance to be identical (μ = σ2 = λ).
When can the Poisson distribution approximate a Binomial distribution?
The Poisson distribution approximates a Binomial distribution when the number of trials is large (n ≥ 20) and the probability of success is small (p ≤ 0.05), using λ = n · p.