Poisson

Poisson Random Variables

Definition[Poisson Random Variable]
A discrete random variable $X$ , with possible values $0,1,2,...$ is called Poisson with parameter $\lambda > 0$ if

$\begin{align*} P(X = i) &= \frac{e^{-\lambda \lambda^{i}}}{i!}, & i = 0,1,2,... \end{align*}$

The expectations, variances, and standard deviations of a Poisson random variable are

$\begin{align*} E[X] &= \lambda \\ V[X] &= \lambda \\ \sigma_X &= \sqrt{\lambda} \end{align*}$

Poisson Approximation to Binomial

One use of the Poisson distribution is as follows,
Proposition:
Suppose that the binomial pmf $b(x;n,p)$ , we let $n\rightarrow \infty$ and $p \rightarrow 0$ such that $np \rightarrow \lambda > 0$ . Then $b(x;n,p) \rightarrow p(x;\lambda)$ .
\end{prop}

Remark:
If $p < 0.1$ and $np \leq 10$ , then such an approximation is generally good, if $np > 10$ then it would be better to use a normal distribution, discussed in a later chapter.

Remark:
What this means is that for experiments with a large number of trials and a low probability $p$ , we can use a Poisson distribution. Since if $p$ is is small then $(1 - p)$ is close to one so we have that $np(1 - p) \approx np = \lambda$ . So we can approximate a Binomial random variable by a Poisson random variable.

Poisson Processes

Definition[Poisson Process]
Suppose at some time, $t = 0$ , we begin counting the number of events that occur. Then for each value of $t$ we obtain a number denoted by $N(t)$ , which is the number of events to occur during $[0,t]$ . For each value of $t$ , $N(t)$ is a discrete random variable with the set of possible values $\{0,1,2,...\}$ . To study this process given by $N(t)$ we make the following simple assumptions about the way that the events occur
\begin{itemize}

Stationary: For all $n \geq 0$ , and for any two equal time intervals $\Delta_1, \Delta_2$ , the probability of $n$ events in $\Delta_1$ is equal to the probability of $n$ events in $\Delta_2$ .
Independent Increments: For all $n \geq 0$ , and for any time interval $(t, t+s)$ , the probability of $n$ events in $(t,t+s)$ is independent of how many events have occurred earlier. In particular, suppose that the times $0 \leq t_1 \leq t_2 < ... < t_k$ are given. For $1 \leq i \leq k -1$ , let $A_i$ be the event that $n_i$ events of the process occur in $[t_i,t_{i+1})$ . The independent increments mean that $\set{A_1, A_2,...,A_k}$ is an independent set of events.
Orderliness:The occurrence of two or more events in a very small time interval is practically impossible. This condition is expressed by
$\begin{align*} \lim_{h \rightarrow 0} P(N(h) > 1)/h = 0. \end{align*}$

This implies that as $h \rightarrow 0$ , the probability of two or more events, $P(N(h) > 1)$ , approaches 0 faster than $h$ does. That is, if $h$ is negligible, then $P(N(h) > 1)$ is even more negligible.

We will prove in a later chapter that:
The simultaneous occurrence of two or more events is impossible. Therefore, under the aforementioned properties, events occur one at a time, not in pairs or groups.

Theorem:
If random events occur in time in a way that the above conditions are always satisfied, $N(0) = 0$ , and , for all $t >0, 0 < P(N(t) = 0) < 1$ , then there exists a positive number $\lambda$ such that

$\begin{align*} P(N(t) = n) &= \frac{(\lambda t)^n e^{-\lambda t}}{n!} \end{align*}$

That is, for all $t > 0, N(t)$ is a Poisson random variable with parameter $\lambda t$ . Hence $E\left[ N(t) \right] = \lambda t$ and therefore $\lambda = E\left[ N(1) \right].$