Definition and Basic Properties
Overview
Poisson distribution: discrete probability distribution describing count of events in fixed interval. Events: occur independently, with constant average rate. Suitable for rare event modeling, e.g., calls per minute, decay counts.
Domain and Range
Domain: non-negative integers \(\{0,1,2,\ldots\}\). Range: probabilities between 0 and 1 summing to 1 over domain.
Basic Assumptions
Events are independent. Probability of two or more events in small interval negligible. Rate parameter \(\lambda\) constant over interval.
Probability Mass Function (PMF)
Formula
P(X = k) = (λ^k * e^(-λ)) / k! , k = 0, 1, 2, ... Interpretation
Probability of exactly \(k\) events in interval. Depends solely on parameter \(\lambda\). PMF sums to 1 over all \(k\).
Example Values
| k (events) | P(X=k), λ=3 |
|---|---|
| 0 | 0.0498 |
| 1 | 0.1494 |
| 2 | 0.2240 |
| 3 | 0.2240 |
| 4 | 0.1680 |
Parameters and Interpretation
Rate Parameter \(\lambda\)
Positive real number \(\lambda>0\). Represents expected count of events per interval. Controls shape and scale of distribution.
Interval Specification
Interval: fixed length, time, area, volume, or other measurable domain. \(\lambda\) must correspond to chosen interval scale.
Effect of \(\lambda\) Changes
Small \(\lambda\): distribution skewed right, mass concentrated near zero. Large \(\lambda\): distribution approaches normal shape (by CLT).
Derivation and Limiting Cases
From Binomial Distribution
Poisson as limit of Binomial(n, p) when \(n \to \infty\), \(p \to 0\), with \(np = \lambda\) fixed.
Mathematical Limit
lim_{n→∞, p→0} Binomial(k; n, p) = (λ^k e^{-λ}) / k!where λ = np Relation to Exponential Distribution
Inter-arrival times of Poisson events: i.i.d. exponential with mean \(1/\lambda\). Poisson counts derived from exponential waiting times.
Moments: Expectation and Variance
Expectation
Mean number of events: \(E[X] = \lambda\). Intuitive: average event count per interval.
Variance
Variance equals mean: \(Var(X) = \lambda\). Characteristic property: equidispersion.
Higher Moments
Skewness: \(1/\sqrt{\lambda}\). Kurtosis excess: \(1/\lambda\). Distribution approaches normality as \(\lambda\) increases.
Poisson Process Connection
Definition
Poisson process: counting process with independent increments, stationary rate \(\lambda\), and Poisson distributed counts.
Properties
Increments independent, counts in disjoint intervals Poisson distributed. Memoryless inter-arrival times.
Applications
Modeling random events in time: arrivals, decay, failures. Foundation for queueing theory, reliability, and stochastic processes.
Applications
Telecommunications
Modeling number of phone calls, packet arrivals in fixed time intervals.
Physics
Radioactive decay counts, photon arrivals in quantum optics.
Biology and Medicine
Modeling mutation counts, disease incidence, rare event occurrences.
Insurance and Finance
Modeling claim counts, default events in risk intervals.
Key Properties
Memorylessness
Counts independent over disjoint intervals. Inter-arrival times exponential, memoryless.
Additivity
Sum of independent Poisson(\(\lambda_i\)) variables is Poisson(\(\sum \lambda_i\)). Useful for aggregation.
Mode
Mode: \(\lfloor \lambda \rfloor\) if \(\lambda\) is not integer; two modes if integer (\(\lambda, \lambda-1\)).
Parameter Estimation
Maximum Likelihood Estimation (MLE)
Estimate \(\hat{\lambda}\) as sample mean of observed counts. MLE unbiased and consistent.
Method of Moments
Equate sample mean to \(\lambda\). Identical to MLE due to equal mean and variance.
Confidence Intervals
Exact intervals using chi-square distribution or normal approximation for large samples.
Limitations and Assumptions
Independence Assumption
Events must occur independently; violation leads to poor model fit.
Constant Rate
Poisson assumes fixed \(\lambda\). Non-homogeneous rates require alternative models.
Equidispersion Constraint
Variance equals mean; overdispersion or underdispersion invalidates Poisson assumption.
Worked Examples
Example 1: Call Center
Average calls per hour \(\lambda=5\). Probability of exactly 3 calls:
P(X=3) = (5^3 * e^{-5}) / 3! = (125 * e^{-5}) / 6 ≈ 0.1404 Example 2: Defects in Manufacturing
Defects per meter average \(\lambda=0.2\). Probability of zero defects in 3 meters:
Scale \(\lambda_{3m} = 0.2 * 3 = 0.6\).
P(X=0) = e^{-0.6} = 0.5488 Summary Table of PMF for λ=2
| k | P(X=k) |
|---|---|
| 0 | 0.1353 |
| 1 | 0.2707 |
| 2 | 0.2707 |
| 3 | 0.1804 |
References
- Feller, W. "An Introduction to Probability Theory and Its Applications," Vol. 1, Wiley, 1968, pp. 222-230.
- Ross, S. "Introduction to Probability Models," 11th ed., Academic Press, 2014, pp. 100-110.
- Kingman, J.F.C. "Poisson Processes," Oxford University Press, 1993, pp. 15-40.
- Grimmett, G., Stirzaker, D. "Probability and Random Processes," 3rd ed., Oxford University Press, 2001, pp. 190-200.
- Johnson, N.L., Kemp, A.W., Kotz, S. "Univariate Discrete Distributions," 3rd ed., Wiley, 2005, pp. 69-85.