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Core Definitions

Fundamental characterizations of Poisson processes

Definition 1: Increment Probability

P\{N(t+h) - N(t) = 1\} = \lambda h + o(h)

P\{N(t+h) - N(t) \geq 2\} = o(h)

For small time intervals h, the probability of exactly one event is approximately λh

Definition 2: Increment Distribution

N(t) - N(s) \sim \text{Poisson}(\lambda(t-s))

The increment over time interval (s, t] follows Poisson distribution

Initial Condition

N(0) = 0

Process starts with no events at time t = 0

Core Properties

Fundamental properties and relationships

Mean and Variance

E[N(t)] = \lambda t

\text{Var}[N(t)] = \lambda t

Both mean and variance grow linearly with time

Covariance Function

\text{Cov}[N(s), N(t)] = \lambda \min(s, t)

Covariance depends only on the earlier time point

Autocorrelation Function

R_N(s, t) = \lambda \min(s, t) + \lambda^2 st

Autocorrelation combines covariance and mean product

Key Distributions

Important probability distributions in Poisson processes

Event Occurrence Time W_n

W_n \sim \Gamma(n, \lambda)

f_{W_n}(t) = \frac{\lambda^n t^{n-1}}{(n-1)!}e^{-\lambda t}

E[W_n] = \frac{n}{\lambda}, \quad \text{Var}[W_n] = \frac{n}{\lambda^2}

Time to n-th event follows gamma distribution

Time Interval T_n

T_n \sim \text{Exp}(\lambda)

f_{T_n}(t) = \lambda e^{-\lambda t}

F_{T_n}(t) = 1 - e^{-\lambda t}

Time between consecutive events follows exponential distribution

Conditional Distribution

P\{N(s) = m \mid N(t) = n\} = \binom{n}{m}\left(\frac{s}{t}\right)^m\left(1-\frac{s}{t}\right)^{n-m}

Given total events, distribution over sub-intervals is binomial

Advanced Properties

Synthesis, decomposition, and non-homogeneous processes

Process Synthesis

N(t) = N_1(t) + N_2(t) \sim \text{Poisson}(\lambda_1 + \lambda_2)

Sum of independent Poisson processes is Poisson with summed intensity

Process Decomposition

N_1(t) \sim \text{Poisson}(\lambda p), \quad N_2(t) \sim \text{Poisson}(\lambda(1-p))

Poisson process can be split into independent sub-processes

Non-Homogeneous Process

N(t) - N(s) \sim \text{Poisson}\left(\int_s^t \lambda(u) du\right)

E[N(t)] = \int_0^t \lambda(u) du

For time-varying intensity, use cumulative intensity function

Applications and Examples

Practical applications and computational examples

Service Systems

Customer Arrivals: λ = customers per hour

Probability: P{arrivals in 2 hours = 5}

P\{N(2) = 5\} = \frac{(2\lambda)^5}{5!}e^{-2\lambda}

Waiting Time: Time to 3rd customer

Distribution: W₃ ~ Γ(3, λ)

Communication Systems

Packet Arrivals: λ = packets per second

Inter-arrival: T ~ Exp(λ)

Queue Length: N(t) at time t

Mean: E[N(t)] = λt

Poisson Processes Formulas