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# The Poisson Process ## Definition Let $N(t)$ count the number of events that occur in the interval $[0, t]$. Then the counting process $\{N(t), t \geq 0\}$ is a **Poisson process** with rate $\lambda > 0$ if 1. $N(0) = 0$ 2. Independent increments 3. The number of events in any interval of le...

# The Poisson Process ## Definition Let $N(t)$ count the number of events that occur in the interval $[0, t]$. Then the counting process $\{N(t), t \geq 0\}$ is a **Poisson process** with rate $\lambda > 0$ if 1. $N(0) = 0$ 2. Independent increments 3. The number of events in any interval of length $t$ is Poisson distributed with mean $\lambda t$. That is, for all $s, t > 0$, $$ P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots $$ ## Interarrival Times Let $T_1$ be the time of the first event, and $T_i$ the time between event $i-1$ and event $i$ for $i > 1$. The sequence $\{T_i, i = 1, 2, \dots\}$ are the **interarrival times**. **Theorem:** The interarrival times for a Poisson process are i.i.d. exponential random variables with mean $1/\lambda$. *Proof*: $$ \begin{aligned} P(T_1 > t) &= P(N(t) = 0) = e^{-\lambda t} \frac{(\lambda t)^0}{0!} = e^{-\lambda t} \\ P(T_i > t \mid T_1 = t_1, \dots, T_{i-1} = t_{i-1}) &= P(0 \text{ events in } (t_1 + \dots + t_{i-1}, t_1 + \dots + t_{i-1} + t]) \\ &= P(N(t) = 0) = e^{-\lambda t} \end{aligned} $$ **Memoryless Property:** For all $s, t > 0$, $$ P(T > t + s \mid T > s) = P(T > t) $$ **Proof :** $$ \begin{aligned} P(T > t + s \mid T > s) &= \frac{P(T > t + s, T > s)}{P(T > s)} = \frac{P(T > t + s)}{P(T > s)} \\ &= \frac{e^{-\lambda (t + s)}}{e^{-\lambda s}} = e^{-\lambda t} = P(T > t) \end{aligned} $$ ## Superposition of Independent Poisson Processes Let $N_1(t)$ and $N_2(t)$ be independent Poisson processes with rates $\lambda_1$ and $\lambda_2$, respectively. Then $N(t) = N_1(t) + N_2(t)$ is a Poisson process with rate $\lambda_1 + \lambda_2$.