泊松过程(Poisson process)

A very important application of the Poisson distribution arises in connection with the occurrence of events of a particular type over time. As an example, suppose that starting from a time point that we label $t=0$, we are interested in counting the number of radioactive pulses(放射性脉冲) recorded by a Geiger counter(盖革计数器). We make the following assumptions about the way in which pulses occur:

• 1. There exists a parameter $\alpha > 0$ such that for any short time interval of length $\Delta t$, the probability that exactly one pulse is receive is $\alpha\cdot\Delta t+o(\Delta t)$.
• 2. The probability of more than one pulse being received during $\Delta t$ is $o(\Delta t)$ [which, along with Assumption 1, implies that the probability of no pulses during $\Delta t$ is $1-\alpha\cdot\Delta t-o(\Delta t)$].
• 3. The number of pulses received during the time interval $\Delta t$ is independent of the number received prior to this time interval.
   

Informally, Assumption 1 says that, for a short interval of time, the probability of receiving a single pulse is approximately proportional to the length of the time interval, where $\alpha$ is the constant of proportionality.

Now let $P_k(t)$ denote the probability that $k$ pulses will be received by the counter during any particular time interval of length $t$.

Proposition. $P_k(t)=\frac{e^{-\alpha t}\cdot(\alpha t)}{k!}$, so that the number of pulses during a time interval of length $t$ is a Poisson rv with parameter $\lambda=\alpha t$. The expected number of pulses during any such time interval is then $\alpha t$, so the expected number during a unit interval of time is $\alpha$.
 

References:

The above content in English is copied from the following book:

《Probability and Statistics For Engineering and The Sciences》(Fifth Edtion) P.131
Author: Jay L. Devore

Section 5 of Chapter 3.