Sebaran Poisson

Experiments yielding numerical values of a random variable X, the number of

outcomes occurring during a given time interval or in a specified region, are called

Poisson experiments. The given time interval may be of any length, such as

a minute, a day, a week, a month, or even a year. Hence a Poisson experiment

can generate observations for the random variable X representing the number of

telephone calls per hour received by an office, the number of days school is closed

due to snow during the winter, or the number of postponed games due to rain

during a baseball season. The specified region could be a line segment, an area,

a volume, or perhaps a piece of material. In such instances X might represent

the number of field mice per acre, the number of bacteria in a given culture, or

the number of typing errors per page. A Poisson experiment is derived from the

Poisson process and possesses the following properties:

Properties of Poisson Process

1. The number of outcomes occurring in one time interval or specified region is

independent of the number that occurs in any other disjoint time interval or

region of space. In this way wc say that the Poisson process has no memory.

2. The probability that a single outcome will occur during a very short time

interval or in a small region is proportional to the length of the time interval

or the size of the region and does not depend on the number of outcomes

occurring outside this time interval or region.

3. The probability that more than one outcome will occur in such a short time

interval or fall in such a small region is negligible.

The number X of outcomes occurring during a Poisson experiment is called a

Poisson random variable, and its probability distribution is called the Poisson

distribution. The mean number of outcomes is computed from p = Xt, where

t is the specific "time," "distance," "area," or "volume" of interest. Since its

probabilities depend on A, the rate of occurrence of outcomes, we shall denote

them by the symbol p(x; Xt). The derivation of the formula for p(x; Xt), based on

the three properties of a Poisson process listed above, is beyond the scope of this

book. The following concept is used for computing Poisson probabilities.

Poisson Distribution

The probability distribution of the Poisson random variable X, representing

the number of outcomes occurring in a given time interval or specified region

denoted by t, is P(x;λt) =  ,    x = 0, 1, 2, ...

where A is the average number of outcomes per unit time, distance, area, or volume, and  e=2.71828

for a few selected values of At ranging from 0.1 to 18. We illustrate the use of this

table with the following two examples.

During a laboratory experiment the average number of radioactive particles passing

through a counter in 1 millisecond is 4. What is the probability that 6 particles

enter the counter in a given millisecond?

Solution: Using the Poisson distribution with x = 6 and Xt = 4 and Table A.2, we have

P(6,4) = =  -   

          = 0.8893 – 0.7851

          = 0.1042

Example :

Ten is the average number of oil tankers arriving each day at a certain port city.

The facilities at the port can handle at most 15 tankers per day. What is the

probability that on a given day tankers have to be turned away?

Solution: Let X be the number of tankers arriving each day. Then, using Table A.2, we have

P(X>15) = 1 – P(X= 1) = 0,0487

Like the binomial distribution, the Poisson distribution is used for quality control,

quality assurance, and acceptance sampling. In addition, certain important

continuous distributions used in reliability theory and queuing theory depend on

the Poisson process. Some of these distributions arc discussed and developed in

Chapter 6.

 Teorem

Both the mean and variance of the Poisson distribution p(x; Xt) are Xt.

The proof of this Theorem is found in Appendix A.26.

In Example 5.20, where Xt — 4, we also have a² — 4 and hence a = 2. Using

Chebyshev's theorem, we can state that our random variable has a probability of at

least 3/4 of falling in the interval p±2a = 4± (2)(2), or from 0 to 8. Therefore, we

conclude that at least three-fourths of the time the number of radioactive particles

entering the counter will be anywhere from 0 to 8 during a given millisecond.

The Poisson Distribution as a Limiting Form of the Binomial

It should be evident from the three principles of the Poisson process that the

Poisson distribution relates to the binomial distribution. Although the Poisson

usually finds applications in space and time problems as illustrated by Examples

5.20 and 5.21, it can be viewed as a limiting form of the binomial distribution.

In the case of the binomial, if n is quite large and p is small, the conditions

begin to simulate the continuous space or time region implications of the Poisson

process. The independence among Bernoulli trials in the binomial case is consistent

with property 2 of the Poisson process. Allowing the parameter p to be close to

zero relates to property 3 of the Poisson process. Indeed, if n is large and p is

close to 0, the Poisson distribution can be used, with p = np, to approximate

binomial probabilities. If p is close to 1, we can still use the Poisson distribution

to approximate binomial probabilities by interchanging what we have defined to

be a success and a failure, thereby changing p to a value close to 0.

Teorem

Let X be a binomial random variable with probability distribution b(x;n,p). When n →∞, p → 0, and η =np remains constant, b(x;n,p) —>p(x;.µ).