Poisson Process

A Poisson process with rate $\lambda$ is a counting process $N(t)$ where events arrive randomly in continuous time with the following properties: 1. $N(0) = 0$ 2. Independent increments: counts in non-overlapping intervals are independent 3. $N(t+s) - N(t) \sim \text{Poisson}(\lambda s)$ for any $t, s \ge 0$ The key connections: - Number of events in time $t$: $N(t) \sim \text{Poisson}(\lambda t)$, so $E[N(t)] = \lambda t$ and $\text{Var}(N(t)) = \lambda t$ - Time between events: $\text{Exp}(\lambda)$, with $E[T] = 1/\lambda$ - Time to $k$-th event: $\text{Gamma}(k, \lambda)$ Concrete example: Customers arrive at a shop at rate 5 per hour. What is the probability of seeing exactly 3 customers in 30 minutes? The rate for half an hour is $\lambda t = 5 \times 0.5 = 2.5$: $$P(N = 3) = \frac{e^{-2.5} \cdot 2.5^3}{3!} = \frac{e^{-2.5} \cdot 15.625}{6} \approx 0.214$$ Superposition: Merging two independent Poisson processes with rates $\lambda_1, \lambda_2$ gives a Poisson process with rate $\lambda_1 + \lambda_2$. Thinning: Randomly keeping each event with probability $p$ gives a Poisson process with rate $p\lambda$. When to use: Modeling arrivals, defaults, trades, or any events occurring "randomly" at a constant average rate. The memoryless property of exponential interarrival times makes it the natural starting point for continuous-time models in finance and queueing.