Poisson

Definition

Distribution Type

Discrete probability distribution. Models count of events occurring independently in fixed interval (time, space, volume).

Parameter

Single parameter λ (lambda), average number of events per interval. λ > 0.

Random Variable

Let X = number of events in interval. X ∈ {0, 1, 2, ...}.

Historical Background

Origin

Introduced by Siméon Denis Poisson in 1837. Developed to model rare events in probability theory.

Motivation

Approximate binomial distribution under rare event, large trials limit.

Evolution

Widely adopted in queueing theory, telecommunication, reliability, and stochastic processes.

Properties

Mean and Variance

Mean = λ. Variance = λ. Equal mean and variance characteristic.

Skewness and Kurtosis

Skewness = 1/√λ. Kurtosis excess = 1/λ.

Memorylessness

Not memoryless (unlike exponential distribution).

Support

Non-negative integers: 0, 1, 2, ...

Closed under summation

If X~Poisson(λ1), Y~Poisson(λ2), independent, then X+Y~Poisson(λ1+λ2).

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 with rate λ.

Example Values

Cumulative Distribution Function (CDF)

Definition

CDF F(k) = P(X ≤ k) = Σ (from i=0 to k) P(X = i).

Computation

Sum of PMF values up to k.

Relationship to Gamma Function

Expressible via incomplete gamma function for non-integer sums.

Moment Generating Function (MGF)

Formula

M_X(t) = E[e^(tX)] = exp(λ(e^t - 1))

Use

Derives moments: mean, variance, higher moments.

Characteristic Function

φ_X(t) = exp(λ(e^(it) - 1)), with i = imaginary unit.

Applications

Queueing Theory

Models arrivals: customers, calls, packets per time unit.

Reliability Engineering

Counts failures/events over operational time.

Biology and Medicine

Models mutation occurrences, radioactive decay, rare disease cases.

Telecommunications

Packet arrivals, error counts, signal noise events.

Insurance and Finance

Counts claims, defaults, rare financial events.

Parameter Estimation

Maximum Likelihood Estimation (MLE)

Estimate λ by sample mean: λ̂ = (Σ x_i) / n.

Method of Moments

Equate sample mean and variance to λ.

Bayesian Estimation

Use conjugate prior Gamma(α, β), posterior also Gamma.

Confidence Intervals

Approximate via normal or exact Poisson intervals.

Relation to Other Distributions

Binomial Distribution

Poisson is limit of Binomial(n, p) as n→∞, p→0, np=λ fixed.

Exponential Distribution

Inter-arrival times in Poisson process are exponential.

Gamma Distribution

Waiting time for k events in Poisson process is Gamma(k, rate=λ).

Normal Approximation

Poisson approximates Normal for large λ.

Compound Poisson

Sum of random variables with Poisson-distributed count.

Simulation and Sampling

Direct Sampling

Use inverse transform method on CDF.

Algorithm (Knuth)

initialize k = 0, p = 1, L = e^(-λ)while p > L: k = k + 1 generate uniform u ~ U(0,1) p = p * ureturn k - 1

Software Functions

Available in R, Python (scipy.stats.poisson), MATLAB.

Applications

Simulate queue arrivals, failure counts, event occurrences.

Limitations and Assumptions

Independence

Events must be independent, no interaction.

Stationarity

Rate λ constant over interval.

Rare Events

Best suited for low probability, high number trials.

No Simultaneity

Events occur one at a time.

Approximation Errors

Deviations arise if assumptions violated.

Examples

Telephone Calls

Calls arriving at switchboard: average 5 per minute, P(X=3) computed by PMF.

Radioactive Decay

Count of decays per second from a sample with λ=0.7.

Defects in Manufacturing

Number of defects per meter of fabric, modeled by Poisson.

Traffic Accidents

Crashes at intersection per day, rare and independent events.

Website Hits

Hits per second in low traffic scenarios modeled by Poisson.

References

• Ross, S.M., Introduction to Probability Models, 11th ed., Academic Press, 2014, pp. 75-85.
• Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley, 1968, pp. 156-160.
• Johnson, N.L., Kemp, A.W., and Kotz, S., Univariate Discrete Distributions, 3rd ed., Wiley, 2005, pp. 120-130.
• Kingman, J.F.C., Poisson Processes, Oxford University Press, 1993, pp. 2-20.
• Grimmett, G.R., and Stirzaker, D.R., Probability and Random Processes, 3rd ed., Oxford University Press, 2001, pp. 247-255.