Thinning (Poisson)

Definition

Let $\{N(t),t\ge 0\}$ be a Poisson process of rate $\lambda$.

Suppose each event is independently classified as either type I with probability $p$ or type II with probability $1-p$. Let $N_1(t)$ and $N_2(t)$ denote the number of type I and type II events by time $t$.

This process of splitting the original process into sub-processes is called Thinning.

Interpretation

Imagine a Poisson stream of customers arriving at a store, and each customer is independently male (prob $p$) or female (prob $1-p$). Thinning separates the stream into two independent Poisson streams — one for males, one for females.

Result

By Thinning Theorem:

• $N_1(t)\sim \text{Po}(\lambda pt)$ — Poisson process with rate $\lambda p$
• $N_2(t)\sim \text{Po}(\lambda (1-p)t)$ — Poisson process with rate $\lambda (1-p)$
• $N_1$ and $N_2$ are independent

Generalizes to $k$ types: if events are classified into $k$ types with probabilities $p_1,\dots ,p_k$, then the $k$ resulting processes are independent Poisson processes with rates $\lambda p_i$.

Related

• Poisson Process
• Independence of Thinned Processes