Textbook Applied Probability and Stochastic Processes Second Edition Beichelt Ebook All Chapter PDF
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Mathematics Second
Edition
Stochastic Processes
applications in science, engineering, finance, computer science, and
operations research. It covers the theoretical foundations for modeling
time-dependent random phenomena in these areas and illustrates Processes
applications through the analysis of numerous practical examples.
New to the Second Edition Second Edition
• Completely rewritten part on probability theory—now more than
double in size
• New sections on time series analysis, random walks, branching
processes, and spectral analysis of stationary stochastic
processes
• Comprehensive numerical discussions of examples, which
replace the more theoretically challenging sections
• Additional examples, exercises, and figures
Presenting the material in a reader-friendly, application-oriented
manner, the author draws on his 50 years of experience in the field to
give readers a better understanding of probability theory and stochastic
processes and enable them to use stochastic modeling in their work.
Many exercises allow readers to assess their understanding of the
topics. In addition, the book occasionally describes connections
between probabilistic concepts and corresponding statistical
approaches to facilitate comprehension. Some important proofs
Frank Beichelt
and challenging examples and exercises are also included for more
theoretically interested readers.
Beichelt
K24109
w w w. c rc p r e s s . c o m
Frank Beichelt
University of the Witwatersrand
Johannesburg, South Africa
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
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CONTENTS
PREFACE
SYMBOLS AND ABBREVIATIONS
INTRODUCTION
10 MARTINGALES
10.1 DISCRETE-TIME MARTINGALES 475
10.1.1 Definition and Examples 475
10.1.2 Doob-Type Martingales 479
10.1.3 Martingale Stopping Theorem and Applications 486
10.2 CONTINUOUS-TIME MARTINGALES 489
10.3 EXERCISES 492
11 BROWNIAN MOTION
11.1 INTRODUCTION 495
11.2 PROPERTIES OF THE BROWNIAN MOTION 497
11.3 MULTIDIMENSIONAL AND CONDITIONAL DISTRIBUTIONS 501
11.4 FIRST PASSAGE TIMES 504
11.5 TRANSFORMATIONS OF THE BROWNIAN MOTION 508
11.5.1 Identical Transformations 508
11.5.2 Reflected Brownian Motion 509
11.5.3 Geometric Brownian Motion 510
11.5.4 Ornstein-Uhlenbeck Process 511
11.5.5 Brownian Motion with Drift 512
11.5.5.1 Definitions and First Passage Times 512
11.5.5.2 Application to Option Pricing 516
11.5.5.3 Application to Maintenance 522
11.5.6 Integrated Brownian Motion 524
11.6 EXERCISES 526
REFERENCES 549
INDEX 553
PREFACE TO THE SECOND EDITION
Probability Theory
X, Y, Z random variables
E(X), Var(X) mean (expected) value of X, variance of X
f X (x), F X (x) probability density function, (cumulative probability) distribution
function of X
F Y (y x), f Y (y x) conditional distribution function, density of Y given X = x
X t , F t (x) residual lifetime of a system of age t, distribution function of X t
E(Y x) conditional mean value of Y given X = x
λ(x), Λ(x) failure rate, integrated failure rate (hazard function)
N(μ, σ 2 ) normally distributed random variable (normal distribution) with
mean value µ and variance σ 2
ϕ(x), Φ(x) probability density function, distribution function of a standard
normal random variable N(0, 1)
f X (x 1 , x 2 , ... , x n ) joint probability density function of X = (X 1 , X 2 , ... , X n )
F X (x 1 , x 2 , ... , x n ) joint distribution function of X = (X 1 , X 2 , ... , X n )
Cov(X, Y), ρ(X, Y) covariance, correlation coefficient between X and Y
M(z) z-transform (moment generating function) of a discrete random
variable or of its probability distribution, respectively
Stochastic Processes
{X(t), t ∈ T}, {X t , t ∈ T} continuous-time, discrete-time stochastic process with
parameter space T
Z state space of a stochastic process
f t (x), F t (x) probability density, distribution function of X(t)
f t 1 ,t 2 ,...,t n (x 1 , x 2 , ... , x n ), F t 1 ,t 2 ,...,t n (x 1 , x 2 , ... , x n )
joint density, distribution function of (X(t 1 ), X(t 2 ), ... , X(t n ))
m(t) trend function of a stochastic process
C(s,t) covariance function of a stochastic process
C(τ) covariance function of a stationary stochastic process
C(t), {C(t), t ≥ 0} compound random variable, compound stochastic process
ρ(s,t) correlation function of a stochastic process
{T 1 , T 2 , ...} random point process
{Y 1 , Y 2 , ...} sequence of interarrival times, renewal process
N integer-valued random variable, discrete stopping time
{N(t), t ≥ 0} (random) counting process
N(s, t) increment of a counting process in (s, t]
H(t), H 1 (t) renewal function of an ordinary, delayed renewal processs
A(t) forward recurrence time, point availability
B(t) backward recurrence time
R(t), {R(t), t ≥ 0} risk reserve, risk reserve process
A, A(t) stationary (long-run) availability, point availability
(n)
p ij , p ij one-step, n-step transition probabilities of a homogeneous,
discrete-time Markov chain
p i j (t); q i j , q i transition probabilities; conditional, unconditional transition rates
of a homogeneous, continuous-time Markov chain
{π i ; i ∈ Z} stationary state distribution of a homogeneous Markov chain
π0 extinction probability, vacant probability (sections 8.5, 9.7)
λj , μj birth, death rates
λ, μ, ρ arrival rate, service rate, traffic intensity λ/μ (in queueing models)
μi mean sojourn time of a semi-Markov process in state i
µ drift parameter of a Brownian motion process with drift
W waiting time in a queueing system
L lifetime, cycle length, queue length, continuous stopping time
L(x) first-passage time with regard to level x
L(a,b) first-passage time with regard to level min(a, b)
{B(t), t ≥ 0} Brownian motion (process)
σ2, σ σ 2 = Var(B(1)) variance parameter, volatility
{S(t), t ≥ 0} seasonal component of a time series (section 6.4), standardized
Brownian motion (chapter 11).
{B(t), 0 ≤ t ≤ 1} Brownian bridge
{D(t), t ≥ 0} Brownian motion with drift
M(t) absolute maximum of the Brownian motion (with drift) in [0, t]
M absolute maximum of the Brownian motion (with drift) in [0, ∞)
{U(t), t ≥ 0} Ornstein-Uhlenbeck process, integrated Brownian motion process
ω, w circular frequency, bandwidth
s(ω), S(ω) spectral density, spectral function (chapter 12)
Introduction
Is the world a well-ordered entirety,
or a random mixture,
which nevertheless is called world-order?
Marc Aurel
Random influences or phenomena occur everywhere in nature and social life. Their
consideration is an indispensable requirement for being successful in natural, econ-
omical, social, and engineering sciences. Random influences partially or fully contri-
bute to the variability of parameters like wind velocity, rainfall intensity, electromag-
netic noise levels, fluctuations of share prices, failure time points of technical units,
timely occurrences of births and deaths in biological populations, of earthquakes, or
of arrivals of customers at service centers. Random influences induce random events.
An event is called random if on given conditions it can occur or not. For instance,
the events that during a thunderstorm a certain house will be struck by lightning, a
child will reach adulthood, at least one shooting star appears in a specified time
interval, a production process comes to a standstill for lack of material, a cancer
patient survives chemotherapy by 5 years are random. Border cases of random events
are the deterministic events, namely the certain event and the impossible event. On
given conditions, a deterministic (impossible) event will always (never) occur. For
instance, it is absolutely sure that lead, when heated to a temperature of over
327.5 0 C will become liquid, but that lead during the heating process will turn to
gold is an impossible event. Random is the shape, liquid lead assumes if poured on an
even steel plate, and random is also the occurrence of events which are predicted from
the form of these castings to the future. Even if the reader is not a lottery, card, or
dice player, she/he will be confronted in her/his daily routine with random influences
and must take into account their implications: When your old coffee machine fails
after an unpredictable number of days, you go to the supermarket and pick a new one
from the machines of your favorite brand. At home, when trying to make your first
cup of coffee, you realize that you belong to the few unlucky ones who picked by
chance a faulty machine. A car driver, when estimating the length of the trip to his
destination, has to take into account that his vehicle may start only with delay, that a
traffic jam could slow down the progress, and that scarce parking opportunities may
cause further delay. Also, at the end of a year the overwhelming majority of the car
drivers realize that having taken out a policy has only enriched the insurance compa-
ny. Nevertheless, they will renew their policy because people tend to prefer moderate
regular cost, even if they arise long-term, to the risk of larger unscheduled cost.
Hence it is not surprising that insurance companies belonged to the first institutions
that had a direct practical interest in making use of methods for the quantitative
evaluation of random influences and gave in turn important impulses for the develop-
2 APPLIED PROBABILITY AND STOCHASTIC PROCESSES
ment of such methods. It is the probability theory, which provides the necessary
mathematical tools for their work.
Probability theory deals with the investigation of regularities random events are
subjected to.
The existence of such statistical or stochastic regularities may come as a surprise to
philosophically less educated readers, since at first glance it seems to be paradoxic-
al to combine regularity and randomness. But even without philosophy and without
probability theory, some simple regularities can already be illustrated at this stage:
1) When throwing a fair die once, then one of the integers from 1 to 6 will appear
and no regularity can be observed. But if a die is thrown repeatedly, then the fraction
of throws with outcome 1, say, will tend to 1/6, and with increasing number of throws
this fraction will converge to the value 1/6. (A die is called fair if each integer has
the same chance to appear.)
2) If a specific atom of a radioactive substance is observed, then the time from the
beginning of its observation to its disintegration cannot be predicted with certainty,
i.e., this time is random. On the other hand, one knows the half-life period of a radio-
active substance, i.e., one can predict with absolute certainty after which time from
say originally 10 gram (trillions of atoms) of the substance exactly 5 gram is left.
3) Random influences can also take effect by superimposing purely deterministic
processes. A simple example is the measurement of a physical parameter, e.g., the
temperature. There is nothing random about this parameter when it refers to a spe-
cific location at a specific time. However, when this parameter has to be measured
with sufficiently high accuracy, then, even under always the same measurement
conditions, different measurements will usually show different values. This is, e.g.,
due to the degree of inaccuracy, which is inherent to every measuring method, and to
subjective moments. A statistical regularity in this situation is that with increasing
number of measurements, which are carried out independently and are not biased by
systematic errors, the arithmetic mean of these measurements converges towards the
true temperature.
4) Consider the movement of a tiny particle in a container filled with a liquid. It
moves along zig-zag paths in an apparently chaotic motion. This motion is generated
by the huge number of impacts the particle is exposed to with surrounding molecules
of the fluid. Under average conditions, there are about 10 21 collisions per second
between particle and molecules. Hence, a deterministic approach to modeling the
motion of particles in a fluid is impossible. This movement has to be dealt with as a
random phenomenon. But the pressure within the container generated by the vast
number of impacts of fluid molecules with the sidewalls of the container is constant.
Examples 1 to 4 show the nature of a large class of statistical regularities:
The superposition of a large number of random influences leads under certain
conditions to deterministic phenomena.
INTRODUCTION 3
Scientist n m m/n
Buffon 4040 2048 0.5080
Pearson 12000 6019 0.5016
Pearson 24000 12012 0.5005
Thus, the more frequently a coin is flipped, the more approaches the ratio m/n the
value 1/2 (compare with example 1 above). In view of the large number of flipp-
ings, this principal observation is surely not a random result, but can be confirmed
by all those readers who take pleasure in repeating these experiments. However,
nowadays the experiment 'flipping a coin' many thousand times is done by a comput-
er with a 'virtual coin' in a few seconds. The ratio m/n is called the relative frequency
of the occurrence of the random event 'head appears.'
Already the expositions made so far may have convinced many readers that random
phenomena are not figments of human imagination, but that their existence is object-
ive reality. There have been attempts to deny the existence of random phenomena by
arguing that if all factors and circumstances, which influence the occurrence of an
event are known, then an absolutely sure prediction of its occurrence is possible. In
other words, the protagonists of this thesis consider the creation of the concept of
randomness only as a sign of 'human imperfection.' The young Pierre Simeon
Laplace (1729 − 1827) believed that the world is down to the last detail governed by
deterministic laws. Two of his famous statements concerning this are: 'The curve
described by a simple molecule of air in any gas is regulated in a manner as certain
as the planetary orbits. The only difference between them lies in our ignorance.' And:
'Give me all the necessary data, and I will tell you the exact position of a ball on a
billiard table' (after having been pushed). However, this view has proved futile both
from the philosophical and the practical point of view. Consider, for instance, a
biologist who is interested in the movement of animals in the wilderness. How on
earth is he supposed to be in a position to collect all that information, which would
allow him to predict the movements of only one animal in a given time interval with
absolute accuracy? Or imagine the amount of information you need and the
corresponding software to determine the exact path of a particle, which travels in a
fluid, when there are 10 21 collisions with surrounding molecules per second. It is an
4 APPLIED PROBABILITY AND STOCHASTIC PROCESSES
unrealistic and impossible task to deal with problems like that in a deterministic way.
The physicist Marian von Smoluchowski (1872 − 1917) wrote in a paper published in
1918 that 'all theories are inadequate, which consider randomness as an unknown
partial cause of an event. The chance of the occurrence of an event can only depend
on the conditions, which have influence on the event, but not on the degree of our
knowledge.'
Already at a very early stage of dealing with random phenomena the need arose to
quantify the chance, the degree of certainty, or the likelihood for the occurrence of
random events. This had been done by defining the probability of random events and
by developing methods for its calculation. For now the following explanation is
given: The probability of a random event is a number between 0 and 1. The imposs-
ible event has probability 0, and the certain event has probability 1. The probability
of a random event is the closer to 1, the more frequently it occurs. Thus, if in a long
series of experiments a random event A occurs more frequently than a random event
B, then A has a larger probability than B. In this way, assigning probabilities to
random events allows comparisons with regard to the frequency of their occurrence
under identical conditions. There are other approaches to the definition of probabili-
ty than the classical (frequency) approach, to which this explanation refers. For
beginners the frequency approach is likely the most comprehensible one.
Gamblers, in particular dice gamblers, were likely the first people, who were in need
of methods for comparing the chances of the occurrence of random events, i.e., the
chances of winning or losing. Already in the medieval poem De Vetula of Richard de
Fournival (ca 1200−1250) one can find a detailed discussion about the total number
of possibilities to achieve a certain number, when throwing 3 dice. Geronimo
Cardano (1501 − 1576) determined in his book Liber de Ludo Aleae the number of
possibilities to achieve the total outomes 2, 3, ..,12, when two dice are thrown. For
instance, there are two possibilities to achieve the outcome 3, namely (1,2) and (2,1),
whereas 2 will be only then achieved, when (1,1) occurs. (The notation (i, j) means
that one die shows an i and the other one a j.) Galileo Galilei (1564 − 1642) proved
by analogous reasoning that, when throwing 3 dice, the probability to get the (total)
outcome 10 is larger than the probability to get a 9. The gamblers knew this from
their experience, and they had asked Galilei to find a mathematical proof. The
Chevalier de Méré formulated three problems related to games of chance and asked
the French mathematician Blaise Pascal (1623 − 1662) for solutions:
1) What is more likely, to obtain at least one 6 when throwing a die four times, or in
a series of 24 throwings of two dice to obtain at least once the outcome (6,6)?
2) How many time does one have to throw two dice at least so that the probability to
achieve the outcome (6,6) is larger than 1/2?
3) In a game of chance, two equivalent gamblers need each a certain number of points
to become winners. How is the stake to fairly divide between the gamblers, when for
some reason or other the game has to be prematurely broken off ? (This problem of
the fair division had been already formulated before de Méré , e.g., in the De Vetula.)
INTRODUCTION 5
Pascal sent these problems to Pierre Fermat (1601 − 1665) and both found their
solutions, although by applying different methods. It is generally accepted that this
work of Pascal and Fermat marked the beginning of the development of probability
theory as a mathematical discipline. Their work has been continued by famous
scientists as Christian de Huygens (1629 − 1695), Jakob Bernoulli (1654 − 1705),
Abraham de Moivre (1667 − 1754), Carl Friedrich Gauss (1777 − 1855), and last
but not least by Simeon Denis de Poisson (1781 − 1840). However, probability theory
was out of its infancy only in the thirties of the twentieth century, when the Russian
mathematician Andrej Nikolajewic Kolmogorov (1903 − 1987) found the solution of
one of the famous Hilbert problems, namely to put probability theory as any other
mathematical discipline on an axiomatic foundation.
Nowadays, probability theory together with its applications in science, medicine,
engineering, economy et al. are integrated in the field of stochastics. The linguistic
origin of this term can be found in the Greek word stochastikon. (Originally, this term
denoted the ability of seers to be correct with their forecasts.) Apart from probability
theory, mathematical statistics is the most important part of stochastics. A key subject
of it is to infer by probabilistic methods from a sample taken from a set of interesting
objects, called among else sample space or universe, to parameters or properties of
the sample space (inferential statistics). Let us assume we have a lot of 10 000
electronic units. To obtain information on what percentage of these units is faulty, we
take a sample of 100 units from this lot. In the sample, 4 units are faulty. Of course,
this figure does not imply that there are exactly 400 faulty units in the lot. But
inferential statistics will enable us to construct lower and upper bounds for the
percentage of faulty units in the lot, which limit the 'true percentage' with a given
high probability. Problems like this led to the development of an important part of
mathematical statistics, the statistical quality control. Phenomena, which depend both
on random and deterministic influences, gave rise to the theory of stochastic
processes. For instance, meteorological parameters like temperature and air pressure
are random, but obviously also depend on time and altitude. Fluctuations of share
prices are governed by chance, but are also driven by periods of economic up and
down turns. Electromagnetic noise caused by the sun is random, but also depends on
the periodical variation of the intensity of sunspots.
Stochastic modeling in operations research comprises disciplines like queueing
theory, reliability theory, inventory theory, and decision theory. All of them play an
important role in applications, but also have given many impulses for the theoretical
enhancement of the field of stochastics. Queueing theory provides the theoretical
fundament for the quantitative evaluation and optimization of queueing systems, i.e.,
service systems like workshops, supermarkets, computer networks, filling stations,
car parks, and junctions, but also military defense systems for 'serving' the enemy.
Inventory theory helps with designing warehouses (storerooms) so that they can on
the one hand meet the demand for goods with sufficiently high probability, and on
the other hand keep the costs for storage as small as possible. The key problem with
dimensioning queueing systems and storage capacities is that flows of customers,
6 APPLIED PROBABILITY AND STOCHASTIC PROCESSES
service times, demands, and delivery times of goods after ordering are subject to
random influences. A main problem of reliability theory is the calculation of the
reliability (survival probability, availability) of a system from the reliabilities of its
subsystems or components. Another important subject of reliability theory is model-
ling the aging behavior of technical systems, which incidentally provides tools for
the survival analysis of human beings and other living beings. Chess automats got
their intelligence from the game theory, which arose from the abstraction of games of
chance. But opponents within this theory can also be competing economic blocs or
military enemies. Modern communication would be impossible without information
theory. This theory provides the mathematical foundations for a reliable transmission
of information although signals may be subject to noise at the transmitter, during
transmission, and at the receiver. In order to verify stochastic regularities, nowadays
no scientist needs to manually repeat thousands of experiments. Computers do this
job much more efficiently. They are in a position to virtually replicate the operation
of even highly complex systems, which are subjected to random influences, to any
degree of accuracy. This process is called (Monte Carlo) simulation. More and very
fruitful applications of stochastic (probabilistic) methods exist in fields like physics
(kinetic gas theory, thermodynamics, quantum theory), astronomy (stellar statistics),
biology (genetics, genomics, population dynamic), artificial intelligence (inference
under undertainty), medicine, genomics, agronomy and forestry (design of experi-
ments, yield prediction) as well as in economics (time series analysis) and social
sciences. There is no doubt that probabilistic methods will open more and more
possibilities for applications, which in turn will lead to a further enhancement of the
field of stochastics.
More than 300 hundreds years ago, the famous Swiss mathematician Jakob Bernoulli
proposed in his book Ars Conjectandi the recognition of stochastics as an independ-
ent new science, the subject of which he introduced as follows:
To conjecture about something is to measure its probability: The Art of conjecturing
or the Stochastic Art is therefore defined as the art of measuring as exactly as possi-
ble the probability of things so that in our judgement and actions we always can
choose or follow that which seems to be better, more satisfactory, safer and more
considered.
In line with Bernoulli's proposal, an independent science of stochastics would have
to be characterized by two features:
1) The subject of stochastics is uncertainty caused by randomness and/or ignorance.
2) Its methods, concepts, and language are based on mathematics.
But even now, in the twenty-first century, an independent science of stochastics is
still far away from being officially established. There is, however, a powerful sup-
port for such a move by internationally leading academics; see von Collani (2003).
PART I
Probability Theory
There is no credibility in sciences in which
no mathematical theory can be applied,
and no credibility in fields which have no
connections to mathematics.
Leonardo da Vinci
CHAPTER 1
Random Events and Their Probabilities
1.1 RANDOM EXPERIMENTS
If water is heated up to 100 0 C at an air pressure of 101 325 Pa, then it will inevitab-
ly start boiling. A motionless pendulum, when being pushed, will start swinging. If
ferric sulfate is mixed with hydrochloric acid, then a chemical reaction starts, which
releases hydrogen sulfide. These are examples for experiments with deterministic
outcomes. Under specified conditions they yield an outcome, which had been known
in advance.
Somewhat more complicated is the situation with random experiments or experim-
ents with random outcome. They are characterized by two properties:
1. Repetitions of the experiment, even if carried out under identical conditions, gen-
erally have different outcomes.
2. The possible outcomes of the experiment are known.
Thus, the outcome of a random experiment cannot be predicted with certainty. This
implies that the study of random experiments makes sense only if they can be repeat-
ed sufficiently frequently under identical conditions. Only in this case stochastic or
statistical regularities can be found.
8 APPLIED PROBABILITY AND STOCHASTIC PROCESSES
Let Ω be the set of possible outcomes of a random experiment. This set is called
sample space, space of elementary events, or universe. Examples of random experi-
ments and their respective sample spaces are:
1) Counting the number of traffic accidents a day in a specified area: Ω = {0, 1, ...}.
2) Counting the number of cars in a parking area with maximally 200 parking bays at
a fixed time point: Ω = {0, 1, ..., 200}.
3) Counting the number of shooting stars during a fixed time interval: Ω = {0, 1, ...}.
4) Recording the daily maximum wind velocity at a fixed location: Ω = [0, ∞).
5) Recording the lifetimes technical systems or organisms: Ω = [0, ∞).
6) Determining the number of faulty parts in a set of 1000: Ω = {0, 1, ..., 1000}.
7) Recording the daily maximum fluctuation of a share price: Ω = [0, ∞).
8) The total profit sombody makes with her/his financial investments a year.
This 'profit' can be negative, i.e. any real number can be the outcome: Ω = (−∞, +∞).
9) Predicting the outcome of a wood reserve inventory in a forest stand: Ω = [0, ∞).
10) a) Number of eggs a sea turtle will bury at the beach: Ω = {0, 1, ...}.
b) Will a baby turtle, hatched from such an egg, reach the water? Ω = {0, 1} with
meaning 0: no, 1: yes.
As the examples show, in the context of a random experiment, the term 'experiment'
has a more general meaning than in the customary sense.
A random experiment may also contain a deterministic component. For instance, the
measurement of a physical quantity should ideally yield the exact (deterministic)
parameter value. But in view of random measurement errors and other (subjective)
influences, this ideal case does not materialize. Depending on the degree of accuracy
required, different measurements, even if done under identical conditions, may yield
different values of one and the same parameter (length, temperature, pressure, amper-
age,...).
3) In training, a hunter shoots at a cardboard dummy. Given that he never fails the
dummy, the latter is the sample space Ω, and any possible impact mark at the dum-
my is an elementary event. Crucial subsets to be hit are e.g. 'head' or 'heart.'
Already these three examples illustrate that often not single elementary events are
interesting, but sets of elementary events. Hence it is not surprising that concepts and
results from set theory play a key role in formally establishing probability theory. For
this reason, next the reader will be reminded of some basic concepts of set theory.
Basic Concepts and Notation from Set Theory A set is given by its elements. We
can consider the set of all real numbers, the set of all rational numbers, the set of all
people attending a performance, the set of buffalos in a national park, and so on. A
set is called discrete if it is a finite or a countably infinite set. By definition, a count-
ably infinite set can be written as a sequence. In other words, its elements can be
numbered. If a set is infinite, but not countably infinite, then it is called nondenumer-
able. Nondenumerable sets are for instance the whole real axis, the positive half-axis,
a finite subinterval of the real axis, or a geometric object (area of a circle, target).
Let A and B be two sets. In what follows we assume that all sets A, B, ... considered
are subsets of a 'universal set' Ω . Hence, for any set A, A ⊆ Ω .
A is called a subset of B if each element of A is also an element of B.
Symbol: A ⊆ B.
The complement of B with regard to A contains all those elements of B which are not
element of A.
Symbol: B\A
In particular, A = Ω\A contains all those elements which are not element of A.
The intersection of A and B contains all those elements which belong both to A and B.
Symbol: A ∩ B
The union of A and B contains all those elements which belong to A or B (or to both).
Symbol: A ∪ B
These relations between two sets are illustrated in Figure 1.1 (Venn diagram). The
whole shaded area is A B.
Ω B
A
A∩B
B\A
A\B
Random Events A random event (briefly: event) A is a subset of the set Ω of all
possible outcomes of a random experiment, i.e. A ⊆ Ω.
A random event A is said to have occurred as a result of a random experiment
if the observed outcome ω of this experiment is an element of A: ω ∈ A.
The empty set ∅ is the impossible event since, for not containing any elementary
event, it can never occur. Likewise, Ω is the certain event, since it comprises all pos-
sible outcomes of the random experiment. Thus, there is nothing random about the
events ∅ and Ω. They are actually deterministic events. Even before having complet-
ed a random experiment, we are absolutely sure that Ω will occur and ∅ will not.
Let A and B be two events. Then the set-theoretic operations introduced above can be
interpreted in terms of the occurrence of random events as follows:
A ∩ B is the event that both A and B occur,
A B is the event that A or B (or both) occur,
If A ⊆ B (A is a subset of B), then the occurrence of A implies the occurrence of B.
A\ B is the set of all those elementary events which are elements of A, but not of B.
Thus, A\ B is the event that A occurs, but not B. Note that (see Figure 1.1)
A\ B = A\ (A ∩ B). (1.3)
The event A = Ω\ A is called the complement of A. It consists of all those elementary
events, which are not in A.
Two events A and B are called disjoint or (mutually) exclusive if their joint occur-
rence is impossible, i.e. if A ∩ B = ∅. In this case the occurrence of A implies that B
cannot occur and vice versa. In particular, A and A are disjoint for any event A ⊆ Ω .
Short Terminology
A∩B A and B
A B A or B
A⊆B A implies B, B follows from A
A\B A but not B
A A not
1 RANDOM EVENTS AND THEIR PROBABILITIES 11
Example 1.1 Let us consider the random experiment 'throwing a die' with sample
space Ω = {1, 2, . . ., 6} and the random events A = {2, 3} and B = {3, 4, 6}. Then,
A ∩ B = {3} and A B = {2, 3, 4, 6}. Thus, if a 3 had been thrown, then both the
events A and B have occurred. Hence, A and B are not disjoint. Moreover, A\B = {2},
B\A = {4, 6}, and A = {1, 4, 5, 6}.
Example 1.2 Two dice D 1 and D 2 are thrown. The sample space is
Ω = {(i 1 , i 2 ), i 1 , i 2 = 1, 2, . . ., 6}.
Thus, an elementary event ω consists of two integers indicating the results i 1 and i 2
of D 1 and D 2 , respectively. Let A = {i 1 + i 2 ≤ 3} and B = {i 1 /i 2 = 2}. Then,
A = {(1, 1), (1, 2), (2, 1)}, B = {(2, 1), (4, 2), (6, 3)}.
Hence,
A ∩ B = {(2, 1}}, A B = {(1, 1), (1, 2), (2, 1), (4, 2), (6, 3)}
and A\B = {(1, 1), (1, 2)}.
1.3 PROBABILITY
The aim of this section consists in constructing rules for determining the probabilities
of random events. Such a rule is principally given by a function P on the set E of all
random events A: P = P(A), A ∈ E.
Note that in this context A is an element of the set E so that the notation A ⊆ E would not be
correct. Moreover, not all subsets of Ω need to be random events, i.e., the set E need not
necessarily be the set of all possible subsets of Ω .
The function P assigns to every event A a number P(A), which is its probability. Of
course, the construction of such a function cannot be done arbitrarily. It has to be
done in such a way that some obvious properties are fulfilled. For instance, if A im-
plies the occurrence of the event B, i.e. A ⊆ B, the B occurs more frequently than A
so that the relation P(A) ≤ P(B) should be valid. If in addition the function P has
properties P(∅) = 0 and P(Ω) = 1 , then the probabilities of random events yield
indeed the desired information about their degree of uncertainty: The closer P(A) is
to 0, the more unlikely is the occurrence of A, and the closer P(A) is to 1, the more
likely becomes the occurrence of A.
12 APPLIED PROBABILITY AND STOCHASTIC PROCESSES
To formalize this intuitive approach, let for now P = P(A) be a function on E with
properties
I) P(∅) = 0, P(Ω) = 1, II) If A ⊆ B, then P(A) ≤ P(B).
As a corollary from these two properties we get the following property of P :
III) For any event A, 0 ≤ P(A) ≤ 1.
Example 1.4 When throwing 3 dice, what is more likely, to achieve the total sum 9
(event A 9 ) or the total sum 10 (event A 10 )? The corresponding sample space is
Ω = {(i, j, k), 1 ≤ i, j, k ≤ 6} with n = 6 3 = 216
possible outcomes. The integers 9 and 10 can be represented a as sum of 3 positive
integers in the following ways:
9 = 3 + 3 + 3 = 4 + 3 + 2 = 4 + 4 + 1 = 5 + 2 + 2 = 5 + 3 + 1 = 6 + 2 + 1,
10 = 4 + 3 + 3 = 4 + 4 + 2 = 5 + 3 + 2 = 5 + 4 + 1 = 6 + 2 + 2 = 6 + 3 + 1.
The sum 3+3+3 corresponds to the event A 333 = 'every die shows a 3' = {(3, 3, 3)}.
The sum 4+3+2 corresponds to the event A 432 that one die shows a 4, another die a
3, and the remaining one a 2:
1 RANDOM EVENTS AND THEIR PROBABILITIES 13
A 432 = {(2, 3, 4), (2, 4, 3), (3, 2, 4), (3, 4, 2), (4, 2, 3), (4, 3, 2)}.
Analogously,
A 441 = {(1, 4, 4), (4, 1, 4), (4, 4, 1)}, A 522 = {(2, 2, 5), (2, 5, 2), (5, 5, 2),
A 531 = {(1, 3, 5), (1, 5, 3), (3, 1, 5), (3, 5, 1), (5, 1, 3), (5, 3, 1)},
A 621 = {(1, 2, 6), (1, 6, 2), (2, 1, 6), (2, 6, 1), (6, 1, 2), (6, 2, 1)}.
Corresponding to the given sum representations for 9 and 10, the numbers of favor-
able elementary events belonging to the events A 9 and A 10 , respectively, are
m A = 1 + 6 + 3 + 3 + 6 + 6 = 25, m B = 2 + 3 + 6 + 6 + 3 + 6 = 27.
Hence, the desired probabilities are:
P(A 9 ) = 25/216 = 0.116, P(A 10 ) = 27/216 = 0.125.
The dice gamblers of the Middle Ages could not mathematically prove this result,
but from their experience they knew that P(A 9 ) < P(A 10 ).
Example 1.6 An optimist buys one ticket in a '6 out of 49' lottery and hopes for hit-
ting the jackpot. What are his chances? There are
⎛ 49 ⎞ = 49 ⋅ 48 ⋅ 47 ⋅ 46 ⋅ 45 ⋅ 44 = 13 983 816
⎝ 6 ⎠ 6!
different possibilities to select 6 numbers out of 49. Thus, one has to fill in almost 14
million tickets to make absolutely sure that the winning one is amongst them. It is
m = 1 and n = 13 983 816. Hence, the probability p 6 of having picked the six 'cor-
rect' numbers is
p6 = 1 = 0.0000000715.
13 983 816
The classical definition of probability satisfies the properties P(∅) = 0 and P(Ω) = 1,
since the impossible event ∅ does not contain any elementary events (m = 0) and
the certain event Ω comprises all elementary events (m = n).
Now, let A and B be two events containing m A and m B elementary events, respectiv-
ely. If A ⊆ B, then m A ≤ m B so that P(A) ≤ P(B). If the events A and B are disjoint,
then they have no elementary events in common so that A B contains m A + m B
elementary events. Hence
m +m m m
P(A B) = A n B = nA + nB = P(A) + P(B)
or P(A B) = P(A) + P(B) if A ∩ B = ∅. (1.6)
More generally, if A 1 , A 2 , . . ., A r are pairwise disjoint events, then
P(A 1 A 2 . . . A r ) = P(A 1 ) + P(A 2 ) + . . . + P(A r ), A i ∩ A k = ∅, i ≠ k. (1.7)
Example 1.7 When participating in the lottery '6 out of 49' with one ticket, what is
the probability of the event A to have at least 4 correct numbers?
Let A i be the event of having got i numbers correct. Then,
A = A4 A5 A6.
A 4 , A 5 , and A 6 are pairwise disjoint events. (It is impossible that there are on one
and the same ticket, say, exactly 4 and exactly 5 correct numbers.) Hence,
P(A) = P(A 4 ) + P(A 5 ) + P(A 6 ) .
There are ( 64 ) = 15 possibilities to choose 4 numbers from the 6 'correct' ones. To
each of these 15 choices there are
⎛ 49 − 6 ⎞ = ⎛ 43 ⎞ = 903
⎝ 6−4 ⎠ ⎝ 2 ⎠
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more than the Latin temperament, under the influence of the pagan
revival, could bear with equanimity. The young Italian mind had had
enough of the creed of abstinence, renunciation, and sacrifice; it
panted for enjoyment. The litanies and the agonies of the Church
repelled it; her self-mortifications and self-mystifications revolted it.
The classic love for form was to oust again the Christian veneration
for the spirit. Virgil ceased to be regarded as a heathen prophet of
Christianity. Scholars ceased to scan his pages for predictions of the
advent of Jesus, and began to revel in the charm of his paganism. In
a former generation Dante had found in the poet of Mantua a ghostly
guide to the Hell, Purgatory, and Heaven of Catholicism; the new
school saw in him a mellifluous minstrel of sensuous joys: a singer of
the beauty of flocks and flowers, of the humming bees, of the trilling
birds, of the murmuring rivulets, of the loves of shepherds and
shepherdesses. The muse of Theocritus had risen from her
enchanted sleep of a thousand years and brought back with her the
sanity and the light that were to banish the phantoms and the mists
of the mediaeval hell. Italy celebrated the resurrection of Pan.
Self-abasement was superseded by self-reverence; and
abstinence by temperance. The dignity of the individual, long lost in
the mediaeval worship of authority, was restored; the glorification of
man succeeded to the glorification of the Kingdom of God on earth.
The beauty of the naked human body was once more recognised
and its cult revived. Fecundity and not chastity became the ideal
virtue. And what the poets described in warm, impassioned melody,
the artists of a later day depicted in no less warm and impassioned
colour. Dante’s ethereal love for Beatrice would have been shocked
at Raphael’s Madonna: Madonna the mother; no longer Madonna
the maiden.
Nor was the new cult confined to profane poets, artists, and
scholars. The divines of the Roman Church were also carried away
by it. Rationalism invaded the Vatican, was petted by the priests, and
promulgated from the pulpit. In sermons preached before the Pope
and his cardinals the dogmas of Christianity were blended with the
doctrines of ancient philosophy, and Hebrew theology was identified
with heathen mythology. Christ’s self-sacrifice was compared to that
of Socrates and of other great and good men of antiquity who had
laid down their lives for the sake of truth and the benefit of mankind.
Pontifical documents were couched in pagan phraseology; the
Father and the Son appeared as Jupiter and Apollo; and the Holy
Virgin as Diana, or even as Venus with the child Cupid; while sacred
hymns were solemnly addressed by pious Catholics to the deities of
Olympus. These and other vagaries were seriously indulged in, after
a fashion abundantly grotesque, but none the less instructive. When
pruned of its absurd extravagances and picturesque ineptitudes, this
enthusiasm for paganism can be regarded both as the fruit and as
the cause of an essentially healthy growth. The Italians of the
fifteenth century succeeded where Julian the Apostate had failed in
the fourth; and to that success may be traced all the subsequent
developments of European culture.
How this revolution came about has been explained at great
length by historians: how, partly through Petrarch’s and Boccaccio’s
influence, the nobles and merchant princes of the Italian republics
took the new learning under their generous patronage; how young
Italian pupils repaired to Constantinople to study the language and
literature of ancient Greece at the feet of men to whom that language
was a living mother tongue; how Greek teachers were encouraged to
bring their treasures to Italy; how they were received by a public as
eager to fathom the mysteries of Greek grammar as a modern public
is to fathom the mysteries of a detective story; and how the stream
gradually swelled into the mighty flood that followed on the fall of
Constantine’s city in 1453. But all this was only a period of gestation.
Modern Europe was really born on the day on which an obscure
Dutch chandler made known to the world the marvellous invention
which was to supersede the scribe’s pen, and to draw forth the torch
of knowledge from the monk’s cell, and from the wealthy merchant’s
study to the crowds in the street.
By a coincidence, apparently strange, the century which opened
the prison-gates of the Christian condemned the Jew to a new
dungeon. The age of the revival of learning and of the printing press
is also the age of vigorous persecution of Israel in Italy. The
compulsory attendance of Jews at divine service now began to be
enforced in a manner more rigid at once and more stupid. Officials
posted at the entrance to the church examined the ears of the Jews,
lest the inward flow of the truth should be stemmed by cottonwool.
Other officials, inside the church, were charged with the duty of
preventing the wretched congregation from taking refuge in sleep. A
Bull of Benedict XIII., issued at Valencia in 1415, decrees that at
least three public sermons a year should be inflicted on the Jews,
and prescribes the arguments that are to be employed for their
conversion: proofs of Christ’s Messianic character drawn from the
Prophets and the Talmud, exposure of the errors and vanities of the
latter book, and demonstration of the fact that the destruction of the
Temple and the woes of the Jews are due to the hardness of their
hearts.
In 1442 Pope Eugenius IV., impelled by the son of an apostate
Jew, ordained that the Jews of Rome should keep their doors and
their windows shut during Easter Week. By 1443 the modest annual
sum of 12 gold pieces, originally contributed by the Jews to the
sports in the Roman circus, had grown to 1130 pieces. Nor were the
Romans any longer content with the extortion of money, but they
now insisted on a personal participation of the Jews in the detested
joys of the arena. The descendants of Titus, and of the Romans who
gazed at the savage spectacle of Jewish captives torn to pieces by
wild beasts, or forced to kill one another for the delectation of the
victors, revived the taste of their remote ancestors for sportful
homicide. The fifteenth-century Carnival in Rome opened with a foot-
race, which was in every respect worthy of its pagan prototype of the
first century. Eight Jews were compelled to appear semi-naked, and,
incited by blows and invectives, to cover the whole of the long
course. Some reached the goal exhausted, others dropped dead on
the way. On the same day the secular and religious chiefs of the
Jewish community were obliged to walk at the head of the
procession of Roman Senators across the course, amidst a tempest
of execration and derision on the part of the mob; while the
eccentricities of the Jew and the prejudices of the Gentile found
similar scope for display upon the stage. In the Carnival plays and
farces of Rome the Jew supplied a stock character that never failed
to provoke the contemptuous merriment of the audience.
And yet, even in the middle of the fifteenth century, we find the
Popes, in defiance of their own decrees, employing Jewish
physicians. Nor does the lot of the Jew appear to have grown
unbearable for some time after. Sixtus IV., whose intolerance
towards the Jews of Spain has been recorded in a previous chapter,
died in 1484, and was succeeded by Innocent VIII., a man of many
superstitions and many children, but a feeble and ineffectual pontiff,
the most interesting year of whose reign, to us, is the year of his
death, 1492. In that year, in which the Renaissance reached its
zenith, the Jewish population of Italy was augmented by the influx of
large numbers of refugees from Spain. One party of them landed at
Genoa; and a heart-rending sight they presented, according to an
eye-witness, as they emerged from the hulls of the vessels and
staggered on to the quay: a host of spectres, haggard with famine
and sickness; men with hollow cheeks and deep-sunken eyes;
mothers scarcely able to stand, fondling their famished infants in
their skeleton arms. On that mole the hapless exiles, shivering under
the blasts of the sea, were allowed to tarry for a short time in order to
refit their vessels, and to recruit themselves for further trials. The law
of the Republic forbade Jewish travellers to remain longer than three
days in the country.
The Genoese monks hastened to make spiritual capital out of
the wanderers’ desolate condition: children, starving, were baptized
in return for a morsel of bread. Those who survived want, illness,
and conversion, and finally left the mole of Genoa, were doomed to
fresh distress. Their own co-religionists declined to receive them at
Rome for fear of competition, and attempted to procure a prohibition
of entry from Innocent’s successor by a bribe of one thousand
ducats. The Pope, however, though not remarkable for tenderness of
heart, was so shocked at the supreme barbarity of the exiles’
brethren that he issued a decree banishing the latter from the city.
The Roman Jews, in order to obtain the repeal of the edict, were
obliged to pay two thousand ducats, and to receive the refugees into
the bargain.
Another contingent reached Naples under equally ghastly
conditions. Their voyage from Spain had been a long martyrdom. A
great many, especially the young and the delicately reared, had
succumbed to hunger and to the foul atmosphere of the narrow and
overcrowded vessels. Others had been murdered by the masters of
the ships for the sake of their property, or were forced to sell their
children in order to defray the expenses of the passage. Those who
escaped the terrors of the sea, and reached the two harbours
mentioned, brought with them an infectious disease, derived from
the privations which they had endured. The infection lurked in Genoa
and Naples through the winter; but when Spring came, it burst forth
into a frightful plague, which spread with terrible rapidity, swept off
upwards of twenty thousand souls in the latter city in one year, and
then extended its wasting arms over the whole of the peninsula.
There can be little doubt that the people, who had elsewhere
been made the scapegoats for epidemics with the origin of which
they had nothing to do, would have been subjected to severe
persecution for a visitation which could certainly be traced to their
agency. But it so happened that the attention of the Italians was this
year, and for many years after, absorbed by other calamities.
On Innocent’s death, Alexander VI. had been raised to St.
Peter’s throne, which he strengthened by his own political genius,
adorned by his magnificent liberality to the artistic genius of others,
and disgraced by his monstrous depravity. Under
1494
Alexander’s reign Italy witnessed the invasion of
Charles VIII. of France, an event which inaugurated a period of
turmoil, and turned the country into a battle-ground for foreign
princes. Rome alone escaped the consequences of this deluge. The
Pope, alarmed at the king’s approach, offered terms of peace, which
the French monarch finally accepted. Independence was secured at
the cost of dignity, and Alexander VI. was enabled to steer safely
amid the storms that raged over the rest of the peninsula. He died in
1503, regretted by a few, execrated by most of his contemporaries.
Pius III. reigned for a few months, and was, in his turn, succeeded by
Julius II., who proved himself one of the most energetic, warlike, and
worldly statesmen that had ever wielded St. Peter’s sceptre. He died
in 1513, and in his stead was elected Giovanni de Medici, under the
name of Leo X. Born in 1475, a year after Ariosto, Giovanni was the
second son of Lorenzo de Medici, chief of the Italian Platonists of the
time. In his father’s house and among his father’s friends young
Giovanni heard a great deal more of Pagan poetry and philosophy
than of Christian theology. But while his contemporary, Ariosto,
nourished in a similar school of thought, denounced the rapacity of
the Roman Court and derided the papal pretensions to temporal
power—laughingly dismissing the fabled gift of Constantine the
Great to Pope Silvester to the realms of the moon—Giovanni
devoted his life to the service of a Church whose doctrines he did not
believe, and to her defence against heresies which he did not detest.
His pontificate, accordingly, was distinguished by the elegant
frivolities of a cultured gentleman far more than by the piety of a
clergyman. Leo’s artistic taste and genial sense of the ludicrous were
among his chief virtues; his love of the chase his greatest vice.
Abstemious in his own diet, he delighted in providing for, and
laughing at, the gluttony of others. But Leo’s principal title to the
grateful remembrance of posterity lies in his munificent
encouragement of art and letters. He died in 1521.
Most of these pontiffs, refined, intelligent, and irreligious, in
fighting the reformers fought enemies to their own power, not the
enemies of Christ. While opposing the spirit of rebellion which the
licentiousness of some of them had brought into existence and the
literary culture of others to maturity, they seem to have ignored the
eternal heretics, the Jews. Under their rule Israel enjoyed one of
those Sabbaths of rest which invariably preceded a new reign of
terror. When an academic feud rent the learned world of the
University of Padua into two factions, instead of the philosophical
question under dispute being, after the fashion of the times, settled
at the point of the rapier, it was submitted to the arbitration of a Jew,
the great scholar Elias del Medigo. This worthy, vested in the
professorial robes, addressed the students of Padua and Florence,
and his decision was accepted as final. Lastly, the gulf between Jew
and Gentile in Italy was bridged by a common philosophical faith.
The Italians of the period, in their eager search after truth, often
strayed into strange paths. Many of them, weary of groping their way
amid the darkness of the scholastic wilderness, rashly ran after any
will-of-the-wisp that held out the promise of light and rest. Among
these aberrations from commonsense was the rage for the Hebrew
mysticism of the Cabbala, which found many susceptible disciples
among the literati of Padua and Florence, and led to close and
cordial relations between representatives of the two creeds. The
omniscient youth Count Giovanni Pico de Mirandola, who had been
initiated into the mysteries of the Cabbala by a Jew, maintained that
these mysteries yielded the most effective proof of the divinity of
Christ, and, what is more remarkable still, he had even converted
Pope Sixtus IV. to his way of thinking. Pico de Mirandola placarded
Rome with a list of nine hundred theses, and invited all European
scholars to come to the city at his own expense that they might be
convinced of the infallibility of the Cabbala, while the Pope took great
pains to have the Cabbalistic writings translated into Latin for the
enlightenment of divinity students. Innocent VIII. was far too old-
fashioned to favour new absurdities; and, while he persecuted
witches and magicians in Germany and preached abortive crusades
against the heretics of the West and the infidels of the East, he
prohibited the reading of Pico’s nonsense. But the craze seized Leo
X. and the early Reformers, and not only theologians but also men of
affairs and men of war fell captives to it. Statesmen and soldiers
devoted themselves to the study of Hebrew, in the pathetic belief that
they had at last secured the magic key to universal wisdom.
Contrariwise, many Hebrew Cabbalists, filling high places in the
Synagogue, found in these theosophic hallucinations a proof of the
divine origin of Christianity and openly embraced it. But apart from
mysticism, the genius of the Renaissance overstepped the iron circle
of Judaism. The charm of Hellenism which had in old times attracted
the Jews of Alexandria, once more prevailed against the Hebrew
hatred of Gentile culture. Jewish youths gladly attended the Italian
universities; the philosophy of Aristotle, the elegant Latinity of Cicero
and the subtle criticism of Quintilian met with keen appreciation
among them; and, though painting and sculpture continued to be
regarded with suspicion, we find Italian Rabbis, like their Christian
colleagues, drawing from pagan mythology illustrations for their
sermons, and even paying, in full synagogue, rhetorical homage to
“that holy goddess Diana.”
Thus Jew and Gentile were drawn near to each other by many
intellectual forces. Even theologians succumbed to the mollifying
influence of the new spirit. Too enlightened to persecute, not
sufficiently in earnest to proselytise, they engaged in friendly and
witty arguments with the Jews on the matter of their religion. Pope
Clement VII. even conceived the plan of a Latin
1523–1534
translation of the Old Testament to be brought about by
a collaboration of Jewish and Christian scholars. Under such illusory
auspices was ushered in the century that was to open to the Jews
the blackest chapter in their black history.
CHAPTER XIII
THE GHETTO