伯努利随机过程定义

A Bernoulli process is an iid sequence of Bernoulli random variables. 这里 iid 的含义是：

In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (iid) if each random variable has the same probability distribution as the others and all are mutually independent.

类似于以下几种场景的模型可以使用伯努利过程：

Sequence of lottery wins/losses
Arrivals (each second) to a bank
Arrivals (at each time slot) to a server

伯努利过程具有 Fresh-Start (Memoryless) 属性，this property usually refers to the cases when the distribution of a “waiting time” until a certain event, does not depend on how much time has elapsed already.

伯努利过程中涉及到的相关随机变量的分布

Number of Arrivals and First Arrival Time

Let T be the first arrival time, then $T \sim geometric(p)$
Let $S_n$ be the number of arrivals from time 1 to time n. Then, $S_n \sim binomial(n, p)$

The distribution of busy periods

下面我举个例子：

一个计算机执行2种类型的任务：priority and non-priority. A slot is busy if the computer executes a priority job, and is idle otherwise. We call a string of idle slots , flanked by busy slots（2侧都是 busy slots）, an idle period. Similarly, we call a string of busy slots, flanked by idle slots, a busy period.

A priority job arrives with probability p at the beginning of each slot, independent of other slots, and requires one full slot to execute. A non-priority job is always available and is executed at a given slot if no priority job is available.

上面的例子很明显是一个伯努利随机过程，在这个随机过程中，包含了几个非常重要的随机变量，它们各自的分布为：

1、T, the time index of the first idle slot： $T \sim geometric(1-p)$

2、B, the length of the first busy period： $B \sim geometric(1-p)$

我来解释一下为什么随机变量 B 服从几何公布。首先，busy period 的开端 - 1 和末端 + 1 的 slot 必须都是 idle 的。另外，由于伯努利随机过程具有 Fresh-Start 属性，当我们从 开端 + 1 的位置开始观察时，就是一个全新的伯努利过程，因此我们想看到第一个 idle slot 的出现就算成功了，得到 idle slot 的概率为 1-p. 课件第8页的图会更形象一些。

3、I, the length of the first idle period： $B \sim geometric(p)$

解释与随机变量 B 的分布类似，不多说了！

Arrival Times and Inter-arrival Times

定义：

The time of an arrival is an arrival time. The sequence of arrival times is denoted by $Y_1,Y_2,\cdots$
The difference between arrival times is an inter-arrival time. The sequence of inter-arrival times is denoted by $T_1,T_2,\cdots$

由上面定义可知：

$T_1=Y_1$
$T_k=Y_k-Y_{k-1}, \; k=2,3,\cdots$

结合 Fresh-Start 属性，不难得出 Bernoulli 随机过程的 inter-arrival times 是 iid geometric random variables. 课件第9页的图会更形象一些。知道了这些知识，我来介绍一个重要随机变量的分布， $Y_k$ =time of the kth success/arrival. 如何求解这个随机变量的分布呢？为了解决这个问题，我们需要把 $Y_k=n$ 分解成2个随机变量来看待：

There are k − 1 arrivals before time n
There is an arrival at time n

由于 Fresh-Start 属性，上面2个事件是相互独立的，从而得到下图中的公式，它叫做帕斯卡分布（Pascal）：

帕斯卡分布

参考资料

http://slpl.cse.nsysu.edu.tw/cpchen/courses/probability/arrivalProcess.pdf