Understanding the Poisson Process

Poisson Process

• In a Bernoulli process, the interval $[0, t]$ is divided into $n$ subintervals of length $\Delta t = t/n$.
• Each subinterval has a probability $p$ of one event occurring.
• The probability of multiple events in a subinterval is negligible.
• The number of events $N(t)$ in $[0, t]$ follows a binomial distribution: $P[N(t) = k] = \binom{n}{k}p^k(1-p)^{n-k}$.
• In a Poisson Process, the interval $[0, t]$ is divided into $n$ subintervals of length $\Delta t = t/n$.
• The probability of one event occurring in each subinterval is $\lambda \Delta t$.
• The probability of more than one event occurring in a subinterval is negligible.
• The number of events $N(t)$ occurring in $[0, t]$ has a Poisson distribution: $P[N(t) = k] = \lim_{n \rightarrow \infty} \binom{n}{k}(\lambda \Delta t)^k(1 - \lambda \Delta t)^{n-k} = \frac{(\lambda t)^k}{k!}e^{-\lambda t}$.
• $\lambda$ represents the rate or intensity of the Poisson process.
• A Poisson process with rate $\lambda > 0$ is a counting process ${N(t), t \geq 0}$ with the following properties:
  • $N(0) = 0$
  • Independent increments
  • The number of events in any interval of length $t$ is Poisson distributed with mean $\lambda t$: $P[N(t+s) - N(s) = k] = \frac{(\lambda t)^k}{k!}e^{-\lambda t}$ for all $s, t \geq 0$.

Interarrival Times

• Let $T_1$ be the time of the first event, $T_2$ the time between the first and the second event, and so on. The sequence ${T_n, n = 1, 2,...}$ is called the sequence of interarrival times.
• The interarrival times of a Poisson process are independent and identically distributed exponential random variables with parameter $\lambda$.
  • $P[T_1 > t] = P[N(t) = 0] = e^{-\lambda t}$
  • $P[T_n > t \mid T_1 = t_1,..., T_{n-1} = t_{n-1}] = P[T_n > t] = P[N(t) = 0] = e^{-\lambda t}$
• Memoryless property: A random variable $X$ is memoryless if $P[X > s+t \mid X > t] = P[X>s]$.
• The exponential distribution is memoryless.

Decomposing Poisson Processes

• Each event of a Poisson process with rate $\lambda$ is classified as either type 1 or type 2, independently of other events.
• Each event is classified as type 1 with probability $p$ and type 2 with probability $1-p$.
• $N_1(t)$ and $N_2(t)$ represent the number of type 1 and type 2 events in $[0, t]$, respectively.
• $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda p$ and $\lambda(1-p)$, respectively.

Merging Poisson Processes

• If $N_1(t)$ and $N_2(t)$ are two independent Poisson processes with rates $\lambda_1$ and $\lambda_2$, respectively, and $N(t) = N_1(t) + N_2(t)$, then $N(t)$ is a Poisson process with rate $\lambda = \lambda_1 + \lambda_2$.

GeoGebra Introduction - What is it?

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