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.

The number of requests for assistance received by a towing service(牵引服务/牽引服務/service de remorquage) is a Poisson process with rate $\alpha=4$ per hour.

• (a) Compute the probability that exactly ten requests are received during a particular $2$-hour period.
• (b) If the operators of the towing services take a $30$-min break for lunch, what is the probability that they do not miss any calls for assistance?
• (c) How many calls would you expect during their break?
   

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate $\alpha=8$ per hour, so that the number of arrivals during a time period of $t$ hours is a Poisson rv with parameter $\lambda=8t$.

• (a) What is the probability that exactly $5$ small aircraft arrive during a $1$-hour period? At least $5$? At least 10?
• (b) What are the expected value and standard deviation of the number of small aircraft that arrive during a $90$-min period?
• (c) What is the probability that at least $20$ small aircraft arrive during a $2\frac{1}{2}$-hour period? That at most $10$ arrive during this period?
   

An article in the Los Angeles Times (Dec. 3, 1993) reports that $1$ in $200$ people carry the defective gene that causes inherited colon cancer. In a sample of $1000$ individuals, what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that

• (a) Between $5$ and $8$ (inclusive) carry the gene.
• (b) At least $8$ carry the gene.
   

Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter $\lambda=20$ (suggested in the article "Dynamic Ride Sharing: Theory and Practice", J. of Transp. Engr., 1997:308--312). What is the probability that the number of drivers will

• (a) Be at most $10$ ?
• (b) Exceed $20$ ?
• (c) Between $10$ and $20$, inclusive? Be strictly between $10$ and $20$ ?
• (d) Exceed the mean number by more than two standard deviations?
   

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of $X$=the total number of male children born to the brothers? What is $E(X)$, and how does it compare to the expected number of male children born to each brother?
 

Suppose that $p=P(\text{male birth})=.5$. A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.

• (a) What is the probability that the family has $x$ male children?
• (b) What is the probability that the family has four children?
• (c) What is the probability that the family has at most four children?
• (d) How many male children would you expect this family to have? How many children would you expect this family to have?
   

A geologist has collected 10 specimens(标本/標本) of basaltic rock(玄武岩岩石) and 10 specimens of granite(花岗岩/花崗岩). The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis.

• (a) What is the pmf of the number of granite specimens selected for analysis?
• (b) What is the probability that all specimens of one of the two types of rock are selected for analysis?
• (c) What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?
   

Each of 12 refrigerators of a certain type has been returned to a distributor because of the presence of a high-pitched oscillating noise(高音调振荡噪声/高音調振盪雜訊/高音振動ノイズ) when the refrigerator is running. Suppose that 5 of these 12 have defective compressors and the other 7 have less serious problems. If they are examined in random order, let $X$=the number among the first 6 examined that have a defective compressor. Compute the following:

• (a) $P(X=1)$
• (b) $P(X\geqslant 4)$
• (c) $P(1\leqslant X\leqslant 3)$
   

The negative binomial rv and distribution are based on an experiment satisfying the following conditions:

1. The experiment consists of a sequence of independent trials.
2. Each trial can result in either a success ($S$) or a failure ($F$).
3. The probability of success is constant from trial to trial, so $P(S\ \text{on trial}\ i)=p$ for $i=1,2,3,\ldots$
4. The experiment continues (trials are performed) until a total of $r$ successes have been observed, where $r$ is a specified positive integer.

The random variable of interest is

$X$=the number of failures that precede the $r$th success; 

$X$ is called a negative binomial random variable because, in contrast to the binomial rv, the number of success is fixed and the number of trials is random.

Possible values of $X$ are $0,1,2,\ldots$. Let $nb(x;r,p)$ denote the pmf of $X$. The event $\{X=x\}$ is equivalent to $\{r-1$ $S$'s in the first $(x+r-1)$ trials and an $S$ on the $(x+r)$th trial $\}$.

\[
\begin{split}
nb(x;r,p)&=P(X=x)\\
&=P(r-1\ S\text{'s}\ \text{on the first}\ x+r-1\ \text{trials})\cdot P(S)\\
&=\binom{x+r-1}{r-1}p^{r-1}(1-p)^x\cdot p\\
&=\binom{x+r-1}{r-1}p^{r}(1-p)^x
\end{split}
\]

Thus we have

Prop. The pmf of the negative binomial rv $X$ with parameters $r=$number of $S$'s and $p=P(S)$ is

\[
nb(x;r,p)=\binom{x+r-1}{r-1}p^{r}(1-p)^x,\quad x=0,1,2,\ldots
\]

In some sources, the negative binomial rv is taken to be the number of trials $X+r$ rather than the number of failures.

In the special case $r=1$, the pmf is

\[
nb(x;1,p)=(1-p)^x p,\quad x=0,1,2,\ldots\tag{*}
\]

Both $X$=number of $F$'s and $Y$=number of trials $(=1+X)$ are referred to in the literatures as geometric random variables(几何随机变量), and the pmf (*) is called the geometric distribution（几何分布）.

Prop. If $X$ is a negative binomial rv with pmf $nb(x;r,p)$, then

\[
E(X)=\frac{r(1-p)}{p},\quad V(X)=\frac{r(1-p)}{p^2}
\]

Finally, by expanding the binomial coefficient in front of $p^r(1-p)^x$ and doing some cancellation, it can be seen that $nb(x;r,p)$ is well defined even when $r$ is not an integer. This generalized negative binomial distribution has been found to fit observed data quite well in a wide variety of applications.

Remark:

负二项分布也称帕斯卡分布（巴斯卡分布）

References:

The above content is copied from the following:

Proposition in section 5 of Chapter 3, BOOK:

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