Properties of Poisson Processes

• $N(s)$ is a Poisson distributed random variable with mean $\lambda s$.
• $N(t + s) - N(s)$ is also a Poisson process and is independent of $N(r)$ for $0\leq r < s$.
  • Known as the “restart property”. $N(t + s) - N(s) \sim \mathrm{Poisson}(\lambda t)$
• $N(s)$ has independent increments.
  • If $s_0 < s_1 < s_2 < \, … \, < s_n$, then $N(s_1) - N(s_0) \perp N(s_2) - N(s_1) \perp ...$
  • $N(s_1) - N(s_0) \sim \mathrm{Poisson}(\lambda (s_1 - s_0))$

If $\{ N(s), s \leq 0 \}$ is a Poisson process with rate $\lambda$,

Property 1: $N(0) = 0$

Property 2: $N(s+t) - N(s) \sim \mathrm{Poisson}(\lambda t)$

Property 3: $N(s)$ ****has independent increments

Thinning a Poisson Process

Let $N(s)$ with rate $\lambda$.

N(s)
|--------|--------|-------|---|--------|--------------------...
         T1       T2     T3   T4      T5                    t

N1(s) (with probability p, like a coin toss)
|--------|----------------|------------|--------------------...
         T1              T3           T5                    t

N2(s) (with probability 1-p)
|-----------------|-----------|-----------------------------...
                  T2          T4                            t

$N_1(s)$ has rate $\lambda p$ and $N_2(s)$ has rate $\lambda(1-p)$.

Note that $N_1(s)$ is independent from $N_2(s)$.

Superposition

Let $N_1(s)$ and $N_2(s)$ is independent with rates $\lambda_1$ and $\lambda_2$. $N(s) = N_1(s) + N_2(s)$ is also a Poisson process with rate $\lambda_1 + \lambda_2$.

Poisson Process Questions

1. What’s the distribution of:
   1. $t_1$: $\mathrm{exp}(\lambda)$
   2. $T_4$: $\mathrm{\Gamma}(4, \lambda)$
   3. $T_1$: $\mathrm{exp}(\lambda)$
   4. $t_4 | t_1$: $\mathrm{exp}(\lambda)$ (they’re independent)
   5. $N(s)$: $\mathrm{Poisson}(\lambda s)$
   6. $N(s+\tau)$: $\mathrm{Poisson}(\lambda (s+\tau))$
2. Are the following statements true?
   1. $t_1 \perp t_2$: True
   2. $N(s+\tau) \perp N(s)$: False
   3. $T_2 \perp T_3$: False
   4. $N(s+\tau) - N(s) \perp N(s)$: True

Inhomogeneous Poisson Processes