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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