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# The Poisson Process ## Introduction The Poisson process is a model for events occurring randomly in time. **Example:** - Customers arriving at a store - Flaws in a roll of tape - Incoming phone calls at a call center - Radioactive decay ## Definition Let $N(t)$ be the number of events occurr...

# The Poisson Process ## Introduction The Poisson process is a model for events occurring randomly in time. **Example:** - Customers arriving at a store - Flaws in a roll of tape - Incoming phone calls at a call center - Radioactive decay ## Definition Let $N(t)$ be the number of events occurring in $[0, t]$. $\{N(t), t \geq 0\}$ is a Poisson process with rate $\lambda > 0$ if: 1. $N(0) = 0$ 2. Independent increments: The number of events in disjoint intervals are independent. 3. The number of events in an interval of length $t$ follows a Poisson distribution with mean $\lambda t$. $$ P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots $$ ## Properties * $N(t)$ has independent increments * $N(t)$ has stationary increments * $E[N(t)] = \lambda t$ * $Var[N(t)] = \lambda t$ ## Interarrival Times Let $T_i$ be the time between the $(i-1)^{th}$ and $i^{th}$ event. Then $T_1, T_2, \dots$ are independent and exponentially distributed with mean $1/\lambda$. $$ P(T_i > t) = e^{-\lambda t} $$ ## Memoryless Property The exponential distribution is memoryless: $$ P(T > t+s \mid T > s) = P(T>t) $$ ## Example Suppose customers arrive at a store according to a Poisson process with rate $\lambda = 10$ customers per hour. 1. What is the probability that no customers arrive in the first 15 minutes? $$ P(N(0.25) = 0) = e^{-10 \cdot 0.25} \frac{(10 \cdot 0.25)^0}{0!} = e^{-2.5} \approx 0.082 $$ 2. What is the probability that at least 2 customers arrive in the first 15 minutes? $$ P(N(0.25) \geq 2) = 1 - P(N(0.25) = 0) - P(N(0.25) = 1) = 1 - e^{-2.5} - e^{-2.5} \cdot 2.5 \approx 0.713 $$ 3. What is the expected time until the first customer arrives? $$ E[T_1] = \frac{1}{\lambda} = \frac{1}{10} \text{ hours} = 6 \text{ minutes} $$