Poisson Process

Definition

Let $\{N(t),t\ge 0\}$ be a counting process.

Then $\{N(t),t\ge 0\}$ is a Poisson Process of rate $\lambda >0$ if:

1. $N(0)=0$
2. The process has independent increments
3. $P(N(t+h)-N(t)=1)=\lambda h+o(h)$
4. $P(N(t+h)-N(t)\ge 2)=o(h)$

Interpretation

A Poisson process models events occurring randomly in continuous time at a constant average rate $\lambda$. Conditions 3 and 4 say: in a tiny interval $h$, the chance of exactly one event is proportional to $h$, and the chance of two or more is negligible.

Equivalent Definition

The counting process $\{N(t),t\ge 0\}$ is a Poisson process of rate $\lambda >0$ if:

1. $N(0)=0$
2. The process has independent increments
3. $N(t+s)-N(s)\sim \text{Po}(\lambda t)$ for all $s,t\ge 0$

Axioms

For infinitesimal $h$:

• $P(\text{1 event in }[t,t+h])=\lambda h+o(h)$
• $P(\text{0 events in }[t,t+h])=1-\lambda h+o(h)$
• $P(\ge 2\text{ events in }[t,t+h])=o(h)$

Where $f(h)=o(h)$ means ${\lim }_{h\to 0}\frac{f(h)}{h}=0$.

Properties

1. $N(t)\sim \text{Po}(\lambda t)$ with $E[N(t)]=\lambda t$
2. Time of first event $T_1\sim \text{Exp}(\lambda )$
3. Inter-arrival times $T_n\stackrel{\text{i.i.d.}}{\sim }\text{Exp}(\lambda )$
4. Waiting time $W_n\sim \text{Gamma}(n,\lambda )$

Related

• Counting Process
• Inter-arrival Times
• Waiting Times
• Thinning
• Nonhomogeneous Poisson Process
• Compound Poisson Process