Exponential Distribution and Poisson Processes

Summary of notes from lectures by Sheldon Jacobson for the course IE 410 at UIUC in Fall 2021.

The exponential distribution

• PDF, CDF, E(X), Var(X).
• Memorylessness.
• Definition: hazard rate.
• CDF from hazard rate.
• Sum of n exponential random variables.
• PDF of a gamma variable.
• Theorem: P(X_1 < X_2) = λ_1/(λ_1 + λ_2), where X_i ~ Exponential(λ_i) and X_1 and X_2 are independent.
• Example: Taxi arrival.
• Theorem: Probability distributions of min_i X_i and max_i X_i, where X_i ~ Exponential(λ_i).
• Theorem: P(X_1 = min(X_1, X_2, ..., X_n)) = λ_1/(λ_1 + λ_2 + ... + λ_n), where X_i ~ Exponential(λ_i) and all X_i are independent.

Poisson Process

• Definition: counting process.
• Definition: Independent increments, stationary increments.
• Definitions of Poisson process and their equivalence.
• Theorem: P(N(s)=k | N(t)=n) = C(n,k) (s/t)^k (1-s/t)^(n-k).
• Definition: Inter-arrival times and arrival times of a counting process.
• Theorem: Inter-arrival times of a poisson process are IID exponential(λ).
• Theorem: Sum of n exp(λ) variables is Gamma(n, λ).

Decomposition of Poisson process

• Theorem: Suppose each event receives a label i with probability p_i, and there are k distinct labels. Let N_i(t) be the number of events labeled i in time t. Then N_i is a poisson process with parameter λp_i. Also, for every t, all N_i(t) are independent. Also, conditional on N(t)=n, we get that N_i(t) ~ Binomial(n, p_i).
• Theorem: Let {N_i: i ∈ [k]} be independent poisson processes, where N_i has parameter λ_i. Let N(t) = N_1(t) + ... + N_k(t). Then N is a poisson process with parameter λ_1 + ... + λ_k.

Coupon collector problem

Competing poisson processes

• If there are two poisson processes, what is the probability of seeing m events of the first before n events of the second?

Conditional on n events, arrival times are distributed as order statistics of uniform distribution

Inhomogeneous poisson process

• Definition
• Theorem: An inhomogeneous Poisson process can be reformulated as a random sample from a (homogeneous) Poisson proces.

Compound poisson process