11.3 Poisson processes

Poisson processes are crucial in modeling random events over time or space. They're defined by key properties like stationarity, independence, and orderliness, with events following a Poisson distribution. Understanding these processes is essential for analyzing various real-world phenomena.

Poisson processes connect to exponential distributions through interarrival times. This relationship is fundamental in probability theory, linking discrete event occurrences to continuous time intervals. Applications range from customer arrivals to equipment failures, making Poisson processes a versatile tool in many fields.

Properties of Poisson Processes

Fundamental Characteristics

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• Poisson process models occurrence of random events over time or space characterized by rate parameter λ
• Stationarity independence of increments and orderliness (no simultaneous events) define key properties
• Number of events in fixed interval follows Poisson distribution with mean λt (t represents interval length)
• Interarrival times between events distributed exponentially and independently
• Memoryless property ensures event occurrence probability independent of time since last event
• Superposition property combines multiple independent Poisson processes into single process (rate equals sum of individual rates)
• Thinning property creates new Poisson process with reduced rate by randomly selecting events

Mathematical Foundations

• Rate parameter λ measures average number of events per unit time or space
• Orderliness property ensures probability of multiple events occurring simultaneously approaches zero
• Independence of increments means events in non-overlapping intervals are statistically independent
• Memoryless property expressed mathematically as P(T > s + t | T > s) = P(T > t) for any s t ≥ 0
• Superposition of n independent Poisson processes with rates λ1 λ2 ... λn results in new process with rate λ = λ1 + λ2 + ... + λn
• Thinning process with probability p creates new Poisson process with rate pλ

Probability Distribution of Poisson Processes

Poisson Distribution Formulas

• Probability of exactly k events in fixed interval t given by Poisson distribution formula $P(X = k) = \frac{(λt)^k e^{-λt}}{k!}$
• Mean and variance of number of events both equal λt
• Moment generating function $M(t) = e^{λ(e^t - 1)}$
• Probability generating function $G(z) = e^{λ(z - 1)}$
• Characteristic function $φ(t) = e^{λ(e^{it} - 1)}$
• Cumulative distribution function expressed using incomplete gamma function
• Limiting distribution approximates normal distribution as λt approaches infinity (mean and variance both λt)

Statistical Properties and Applications

• Poisson distribution models rare events in large populations (insurance claims earthquakes)
• Skewness of Poisson distribution equals $\frac{1}{\sqrt{λt}}$ indicating right-skewed nature for small λt
• Kurtosis equals $\frac{1}{λt}$ measuring peakedness relative to normal distribution
• Law of small numbers states binomial distribution approaches Poisson as n increases and p decreases while np remains constant
• Poisson distribution serves as approximation for binomial distribution when n is large and p is small
• Central Limit Theorem applies to Poisson distribution allowing normal approximation for large λt

Modeling Real-World Phenomena with Poisson Processes

Applications in Business and Engineering

• Customer arrivals in queuing theory modeled for various settings (call centers emergency rooms retail stores)
• Reliability engineering uses Poisson processes to model equipment failures or breakdowns over time
• Spatial Poisson process models distribution of objects in multi-dimensional space (trees in forest stars in galaxy)
• Finance applications include modeling arrival of trading orders or rare events (market crashes)
• Telecommunications employs Poisson processes to model data packet or phone call arrivals in networks

Advanced Poisson Process Variants

• Non-homogeneous Poisson process with time-varying rate λ(t) models phenomena with varying intensity (traffic flow seasonal demand)
• Compound Poisson processes associate random variables with each event modeling scenarios (insurance claim amounts rainfall quantities)
• Marked Poisson processes attach additional information to each event (customer type in queue severity of equipment failure)
• Spatial Poisson processes extend to higher dimensions modeling event distributions in 3D space (cosmic ray impacts)
• Cox processes (doubly stochastic Poisson processes) introduce randomness to rate parameter λ itself (modeling events with uncertain rates)

Poisson Processes vs Exponential Distribution

Interrelationship and Properties

• Interarrival times in Poisson process follow exponential distribution with rate parameter λ
• Exponential distribution probability density function $f(x) = λe^{-λx}$ for x ≥ 0
• Cumulative distribution function of exponential distribution $F(x) = 1 - e^{-λx}$ for x ≥ 0
• Mean and standard deviation of exponential distribution both equal $\frac{1}{λ}$
• Memoryless property of exponential distribution corresponds to Poisson process memoryless property
• Waiting time distributions in Poisson processes derived from relationship between Poisson and exponential distributions
• Sum of n independent exponential random variables with rate λ follows Erlang distribution (related to time until nth event in Poisson process)

Applications and Extensions

• Exponential distribution models time between events in Poisson process (time between customer arrivals radioactive decay intervals)
• Survival analysis uses exponential distribution to model lifetimes of components or organisms
• Competing risks models combine multiple exponential distributions to analyze systems with various failure modes
• Phase-type distributions generalize exponential distribution modeling more complex waiting time scenarios
• Residual life in reliability theory leverages memoryless property of exponential distribution
• Continuous-time Markov chains use exponential distribution to model state transition times
• Queueing theory extensively employs exponential distribution in modeling service times and interarrival times (M/M/1 queue)