Gregersen E. (editor) The Britannica Guide to Statistics and Probability

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7 The Britannica Guide to Statistics and Probability 7

110

several servers and different classes of customers, who are

served according to certain priorities. In most cases, it is

impossible to give a mathematical analysis of the system,

which must be simulated on a computer in order to obtain

numerical results. The insights gained from theoretical

analysis of simple cases can be helpful in performing these

simulations. Queuing theory had its origins in attempts

to understand trafﬁc in telephone systems. Present-day

research is stimulated, among other things, by problems

associated with multiple-user computer systems.

Reﬂecting barriers arise in other problems as well.

For example, if B(t) denotes Brownian motion, then

X(t) = B(t) + ct is called Brownian motion with drift c. This

model is appropriate for Brownian motion of a particle

under the inﬂuence of a constant force ﬁeld such as grav-

ity. One can add a reﬂecting barrier at 0 to account for

reﬂections of the Brownian particle off the bottom of its

container. The result is a model for sedimentation, which

for c < 0 in the steady state as t → ∞ gives a statistical deri-

vation of the law of pressure as a function of depth in an

isothermal atmosphere. Just as ordinary Brownian motion

can be obtained as the limit of a rescaled random walk as

the number of steps becomes large and the size of individ-

ual steps small, Brownian motion with a reﬂecting barrier

at 0 can be obtained as the limit of a rescaled random walk

with reﬂection at 0. In this way, Brownian motion with

a reﬂecting barrier plays a role in the analysis of queuing

systems. In fact, in modern probability theory one of the

most important uses of Brownian motion and other diffu-

sion processes is as approximations to more complicated

stochastic processes. The exact mathematical description

of these approximations gives remarkable generalizations of

the central limit theorem from sequences of random vari-

ables to sequences of random functions.

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Probability Theory 7

Insurance Risk Theory

The ruin problem of insurance risk theory is closely

related to the problem of gambler’s ruin described earlier

and, rather surprisingly, to the single-server queue as well.

Suppose the amount of capital at time t in one portfolio of

an insurance company is denoted by X(t). Initially

X(0) = x > 0. During each unit of time, the portfolio

receives an amount c > 0 in premiums. At random times

claims are made against the insurance company, which

must pay the amount V

> 0 to settle the nth claim. If N(t)

denotes the number of claims made in time t, then

provided that this quantity has been positive at all earlier

times s < t. At the ﬁrst time X(t) becomes negative, however,

the portfolio is ruined. A principal problem of insurance

risk theory is to ﬁnd the probability of ultimate ruin. If one

imagines that the problem of gambler’s ruin is modiﬁed so

that Peter’s opponent has an inﬁnite amount of capital and

can never be ruined, then the probability that Peter is ulti-

mately ruined is similar to the ruin probability of insurance

risk theory. In fact, with the artiﬁcial assumptions that (i)

c = 1, (ii) time proceeds by discrete units, say t = 1, 2,…, (iii) V

is identically equal to 2 for all n, and (iv) at each time t a

claim occurs with probability p or does not occur with prob-

ability q independently of what occurs at other times, then

the process X(t) is the same stochastic process as Peter’s

fortune, which is absorbed if it ever reaches the state 0. The

probability of Peter’s ultimate ruin against an inﬁnitely rich

adversary is easily obtained by taking the limit of equation

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112

(6) as m → ∞. The answer is (q/p)

if p > q (i.e., the game is

favourable to Peter) and 1 if p ≤ q. More interesting assump-

tions for the insurance risk problem are that the number

of claims N(t) is a Poisson process and the sizes of the

claims V

, V

,… are independent, identically distributed

positive random variables. Rather surprisingly, under

these assumptions the probability of ultimate ruin as a

function of the initial fortune x is exactly the same as the

stationary probability that the waiting time in the single-

server queue with Poisson input exceeds x. Unfortunately,

neither problem is easy to solve exactly, however, there is

an excellent approximate solution originally derived by

the Swedish mathematician Harald Cramér.

Martingale Theory

As a ﬁnal example, it seems appropriate to mention one

of the dominant ideas of modern probability theory,

which at the same time springs directly from the rela-

tion of probability to games of chance. Suppose that X

,… is any stochastic process and, for each n = 0, 1,…,

= f

,…, X

) is a (Borel-measurable) function of the

indicated observations. The new stochastic process f

called a martingale if E(f

,…, X

n − 1

) = f

n − 1

for every value

of n > 0 and all values of X

,…, X

n − 1

. If the sequence of

Xs are outcomes in successive trials of a game of chance

and f

is the fortune of a gambler after the nth trial, then

the martingale condition says that the game is absolutely

fair in the sense that, no matter what the past history of

the game, the gambler’s conditional expected fortune

after one more trial is exactly equal to his present for-

tune. For example, let X

= x, and for n ≥ 1 let X

equal

1 or −1 according as a coin having probability p of heads

and q = 1 − p of tails turns up heads or tails on the nth toss.

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Probability Theory 7

Let S

= X

+⋯+ X

. Then f

= S

− n ( p − q ) and f

= ( q / p )

S n

are

martingales. One basic result of martingale theory is that,

if the gambler is free to quit the game at any time using

any strategy whatsoever, provided only that this strategy

does not foresee the future, then the game remains fair.

This means that, if N denotes the stopping time at which

the gambler’s strategy tells him to quit the game, so that

his ﬁ nal fortune is f

, then

Strictly speaking, this result is not true without some

additional conditions that must be veriﬁ ed for any partic-

ular application. To see how efﬁ ciently it works, consider

once again the problem of gambler’s ruin and let N be the

ﬁ rst value of n such that S

= 0 or m. For example, N denotes

the random time at which ruin ﬁ rst occurs and the game

ends. In the case p = 1/2, application of

to the martingale f

= S

, together with the observation

that f

= either 0 or m , yields the equalities x = f

= E ( f

| f

x ) = m [1 −

( x )], which can be immediately solved to give

the answer in equation (6)

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114

For p ≠ 1/2, one uses the martingale f

= ( q / p )

S n

and simi-

lar reasoning to obtain

from which the ﬁ rst equation in (6)

easily follows. The expected duration of the game is

obtained by a similar argument.

A particularly beautiful and important result is the

martingale convergence theorem, which implies that a

nonnegative martingale converges with probability 1 as

n → ∞. This means that if a gambler’s successive fortunes

form a (nonnegative) martingale, then they cannot con-

tinue to ﬂ uctuate indeﬁ nitely but must approach some

limiting value.

Basic martingale theory and many of its applications

were developed by the American mathematician Joseph

Leo Doob during the 1940s and ’50s following some ear-

lier results due to Paul Lévy. Subsequently, it has become

one of the most powerful tools available to study stochas-

tic processes.

115

stAtIstICs

CHAPTER 3

tatistics is the science of collecting, analyzing, present-

ing, and interpreting data. Governmental needs for

census data as well as information about a variety of eco-

nomic activities provided much of the early impetus for

the ﬁ eld of statistics. Currently, the need to turn the large

amounts of data available in many applied ﬁ elds into use-

ful information has stimulated both theoretical and

practical developments in statistics.

Data are the facts and ﬁ gures that are collected,

analyzed, and summarized for presentation and interpre-

tation. Data may be classiﬁ ed as either quantitative or

qualitative. Quantitative data measure either how much

or how many of something, and qualitative data provide

labels, or names, for categories of like items. For example,

suppose that a particular study is interested in characteris-

tics such as age, gender, marital status, and annual income

for a sample of 100 individuals. These characteristics

would be called the variables of the study, and data values

for each of the variables would be associated with each

individual. Thus, the data values of 28, male, single, and

$30,000 would be recorded for a 28-year-old single male

with an annual income of $30,000. With 100 individu-

als and 4 variables, the data set would have 100 × 4 = 400

items. In this example, age and annual income are quan-

titative variables; the corresponding data values indicate

how many years and how much money for each individual.

Gender and marital status are qualitative variables. The

labels male and female provide the qualitative data for

gender, and the labels single, married, divorced, and wid-

owed indicate marital status.

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116

Sample survey methods are used to collect data from

observational studies, and experimental design methods

are used to collect data from experimental studies. The

area of descriptive statistics is concerned primarily with

methods of presenting and interpreting data using graphs,

tables, and numerical summaries. Whenever statisticians

use data from a sample (i.e., a subset of the population) to

make statements about a population, they are performing

statistical inference. Estimation and hypothesis testing

are procedures used to make statistical inferences. Fields

such as health care, biology, chemistry, physics, education,

engineering, business, and economics make extensive use

of statistical inference.

Methods of probability were developed initially for the

analysis of gambling games. Probability plays a key role in

statistical inference. It is used to provide measures of the

quality and precision of the inferences. Many methods of

statistical inference are described in this chapter. Some

are used primarily for single-variable studies, whereas oth-

ers, such as regression and correlation analysis, are used to

make inferences about relationships among two or more

variables.

descRipTive sTaTisTics

Descriptive statistics are tabular, graphical, and numerical

summaries of data. The purpose of descriptive statistics is

to facilitate the presentation and interpretation of data.

Most statistical presentations appearing in newspapers

and magazines are descriptive in nature. Univariate meth-

ods of descriptive statistics use data to enhance the

understanding of a single variable. Multivariate methods

focus on using statistics to understand the relationships

among two or more variables. To illustrate methods of

descriptive statistics, the previous example in which data

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Statistics 7

were collected on the age, gender, marital status, and

annual income of 100 individuals will be examined.

Tabular Methods

The most commonly used tabular summary of data for a

single variable is a frequency distribution. A frequency dis-

tribution shows the number of data values in each of

several nonoverlapping classes. Another tabular summary,

called a relative frequency distribution, shows the frac-

tion, or percentage, of data values in each class. The most

common tabular summary of data for two variables is a

cross tabulation, a two-variable analogue of a frequency

distribution.

For a qualitative variable, a frequency distribution

shows the number of data values in each qualitative cate-

gory. For instance, the variable gender has two categories:

male and female. Thus, a frequency distribution for gen-

der would have two nonoverlapping classes to show the

number of males and females. A relative frequency distri-

bution for this variable would show the fraction of

individuals that are male and the fraction of individuals

that are female.

Constructing a frequency distribution for a quantita-

tive variable requires more care in deﬁning the classes and

the division points between adjacent classes. For instance,

if the age data of the example above ranged from 22 to 78

years, then the following six nonoverlapping classes could

be used: 20–29, 30–39, 40–49, 50–59, 60–69, and 70–79. A

frequency distribution would show the number of data

values in each of these classes, and a relative frequency dis-

tribution would show the fraction of data values in each.

A cross tabulation is a two-way table with the rows of

the table representing the classes of one variable and the

columns of the table representing the classes of another

7 The Britannica Guide to Statistics and Probability 7

118

variable. To construct a cross tabulation using the vari-

ables gender and age, gender could be shown with two

rows, male and female, and age could be shown with six

columns corresponding to the age classes 20–29, 30–39,

40–49, 50–59, 60–69, and 70–79. The entry in each cell of

the table would specify the number of data values with the

gender given by the row heading and the age given by the

column heading. Such a cross tabulation could help under-

stand the relationship between gender and age.

Graphical Methods

Many graphical methods are available for describing data.

A bar graph is a graphical device for depicting qualitative

data that have been summarized in a frequency distribu-

tion. Labels for the categories of the qualitative variable

are shown on the horizontal axis of the graph. A bar

If an independent variable is not expressly temporal, a bar graph may be

used to show discrete numerical quantities in relation to each other. Brand

New Images/Stone/Getty Images

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Statistics 7

above each label is constructed such that the height of

each bar is proportional to the number of data values in

the category. A pie chart is another graphical device for

summarizing qualitative data. The size of each slice of

the pie is proportional to the number of data values in the

corresponding class.

A histogram is the most common graphical presenta-

tion of quantitative data that have been summarized in a

frequency distribution. The values of the quantitative

variable are shown on the horizontal axis. A rectangle is

drawn above each class such that the base of the rectangle

is equal to the width of the class interval and its height is

proportional to the number of data values in the class.

Numerical Measures

A variety of numerical measures are used to summarize

data. The proportion, or percentage, of data values in each

category is the primary numerical measure for qualitative

data. The mean, median, mode, percentiles, range, vari-

ance, and standard deviation are the most commonly used

numerical measures for quantitative data. The mean,

often called the average, is computed by adding all the

data values for a variable and dividing the sum by the num-

ber of data values. The mean is a measure of the central

location for the data. The median is another measure of

central location that, unlike the mean, is not affected by

extremely large or extremely small data values. When

determining the median, the data values are ﬁrst ranked in

order from the smallest value to the largest value. If there

is an odd number of data values, then the median is the

middle value. If there is an even number of data values,

then the median is the average of the two middle values.

The third measure of central tendency is the mode, the

data value that occurs with greatest frequency.