Probability and Random Processes (eBook)

Venkatarama Krishnan (Autor)

eBook Download: PDF | EPUB

2015 | 2. Auflage
525 Seiten
Wiley (Verlag)
978-1-119-01190-3 (ISBN)

Infos

The second edition enhanced with new chapters, figures, and appendices to cover the new developments in applied mathematical functions

This book examines the topics of applied mathematical functions to problems that engineers and researchers solve daily in the course of their work. The text covers set theory, combinatorics, random variables, discrete and continuous probability, distribution functions, convergence of random variables, computer generation of random variates, random processes and stationarity concepts with associated autocovariance and cross covariance functions, estimation theory and Wiener and Kalman filtering ending with two applications of probabilistic methods. Probability tables with nine decimal place accuracy and graphical Fourier transform tables are included for quick reference. The author facilitates understanding of probability concepts for both students and practitioners by presenting over 450 carefully detailed figures and illustrations, and over 350 examples with every step explained clearly and some with multiple solutions.

Additional features of the second edition of Probability and Random Processes are:

• Updated chapters with new sections on Newton-Pepys' problem; Pearson, Spearman, and Kendal correlation coefficients; adaptive estimation techniques; birth and death processes; and renewal processes with generalizations
• A new chapter on Probability Modeling in Teletraffic Engineering written by Kavitha Chandra
• An eighth appendix examining the computation of the roots of discrete probability-generating functions

With new material on theory and applications of probability, Probability and Random Processes, Second Edition is a thorough and comprehensive reference for commonly occurring problems in probabilistic methods and their applications.

Venkatarama Krishnan, PhD., is Professor Emeritus in the Department of Electrical and Computer Engineering at the University of Massachusetts at Lowell. He has served as a consultant to the Dynamics Research Corporation, the U.S. Department of Transportation, and Bell Laboratories. Dr. Krishnan's research includes estimation of steady-state queue distribution, tomographic imaging, aerospace, control, communications, and stochastic systems. Dr. Krishnan is a senior member of the IEEE and listed in Who is Who in America.

Textauszug

1
SETS, FIELDS, AND EVENTS

1.1 SET DEFINITIONS

The concept of sets play an important role in probability. We will define a set in the following paragraph.

Definition of Set

A set is a collection of objects called elements. The elements of a set can also be sets. Sets are usually represented by uppercase letters A, and elements are usually represented by lowercase letters a. Thus

(1.1.1)

will mean that the set A contains the elements a1, a2, … , an. Conversely, we can write that ak is an element of A as

(1.1.2)

and ak is not an element of A as

(1.1.3)

A finite set contains a finite number of elements, for example, S = {2,4,6}. Infinite sets will have either countably infinite elements such as A = {x∶x is all positive integers} or uncountably infinite elements such as B = {x∶x is real number ≤ 20}.

Example 1.1.1

The set A of all positive integers less than 7 is written as

Example 1.1.2

The set N of all positive integers is written as

Example 1.1.3

The set R of all real numbers is written as

Example 1.1.4

The set R2 of real numbers x, y is written as

Example 1.1.5

The set C of all real numbers x,y such that x+y ≤ 10 is written as

Venn Diagram

Sets can be represented graphically by means of a Venn diagram. In this case we assume tacitly that S is a universal set under consideration. In Example 1.1.5, the universal set S = {x∶x is all positive integers}. We shall represent the set A in Example 1.1.1 by means of a Venn diagram of Fig. 1.1.1.

Figure 1.1.1

Empty Set

An empty is a set that contains no element. It plays an important role in set theory and is denoted by Ø. The set A = {0} is not an empty set since it contains the element 0.

Cardinality

The number of elements in the set A is called the cardinality of set A, and is denoted by |A|. If it is an infinite set, then the cardinality is ∞.

Example 1.1.6

The cardinality of the set A = {2,4,6} is 3, or |A| = 3. The cardinality of set R = {x∶x is real} is ∞.

Example 1.1.7

The cardinality of the set A = {x∶x is positive integer <7} is |A| = 6.

Example 1.1.8

The cardinality of the set B = {x∶x is a real number <10} is infinity since there are infinite real numbers <10.

Subset

A set B is a subset of A if every element in B is an element of A and is written as B⊆A. B is a proper subset of A if every element of A is not in B and is written as B⊂A.

Equality of Sets

Two sets A and B are equal if B⊆A and A⊆B, that is, if every element of A is contained in B and every element of B is contained in A. In other words, sets A and B contain exactly the same elements. Note that this is different from having the same cardinality, that is, containing the same number of elements.

Example 1.1.9

The set B = {1,3,5} is a proper subset of A = {1,2,3,4,5,6}, whereas the set C = {x∶x is a positive even integer ≤ 6} and the set D = {2,4,6} are the same since they contain the same elements. The cardinalities of B, C, and D are 3 and C = D.

We shall now represent the sets A and B and the sets C and D in Example 1.1.9 by means of the Venn diagram of Fig. 1.1.2 on a suitably defined universal set S.

Figure 1.1.2

Power Set

The power set of any set A is the set of all possible subsets of A and is denoted by PS(A). Every power set of any set A must contain the set A itself and the empty set Ø. If n is the cardinality of the set A, then the cardinality of the power set |PS(A)| = 2n.

Example 1.1.10

If the set A = {1,2,3} then PS(A) = {Ø, (1,2,3), (1,2), (2,3), (3,1), (1), (2), (3)}. The cardinality |PS(A)| = 8 = 23.

1.2 SET OPERATIONS

Union

Let A and B be sets belonging to the universal set S. The union of sets A and B is another set C whose elements are those that are in either A or B, and is denoted by A∪B. Where there is no confusion, it will also be represented as A+B:

(1.2.1)

Example 1.2.1

The union of sets A = {1,2,3} and B = {2,3,4,5} is the set C = A∪B = {1,2,3,4,5}.

Intersection

The intersection of the sets A and B is another set C whose elements are the same as those in both A and B and is denoted by A∩B. Where there is no confusion, it will also be represented by AB.

(1.2.2)

Example 1.2.2

The intersection of the sets A and B in Example 1.2.1 is the set C = {2,3} Examples 1.2.1 and 1.2.2 are shown in the Venn diagram of Fig. 1.2.1.

Figure 1.2.1

Mutually Exclusive Sets

Two sets A and B are called mutually exclusive if their intersection is empty. Mutually exclusive sets are also called disjoint.

(1.2.3)

One way to determine whether two sets A and B are mutually exclusive is to check whether set B can occur when set A has already occurred and vice versa. If it cannot, then A and B are mutually exclusive. For example, if a single coin is tossed, the two sets, {heads} and {tails}, are mutually exclusive since {tails} cannot occur when {heads} has already occurred and vice versa.

Independence

We will consider two types of independence. The first is known as functional independence [58].1 Two sets A and B can be called functionally independent if the occurrence of B does not in any way influence the occurrence of A and vice versa. The second one is statistical independence, which is a different concept that will be defined later. As an example, the tossing of a coin is functionally independent of the tossing of a die because they do not depend on each other. However, the tossing of a coin and a die are not mutually exclusive since any one can be tossed irrespective of the other. By the same token, pressure and temperature are not functionally independent because the physics of the problem, namely, Boyle’s law, connects these quantities. They are certainly not mutually exclusive.

Cardinality of Unions and Intersections

We can now ascertain the cardinality of the union of two sets A and B that are not mutually exclusive. The cardinality of the union C = A∪B can be determined as follows. If we add the cardinality |A| to the cardinality |B|, we have added the cardinality of the intersection |A∩B| twice. Hence we have to subtract once the cardinality |A∩B| as shown in Fig. 1.2.2. Or, in other words

(1.2.4a)

In Fig. 1.2.2 the cardinality |A| = 9 and the cardinality |B| = 11 and the cardinality |A∪B| is 11+9−4 = 16.

As a corollary, if sets A and B are mutually exclusive, then the cardinality of the union is the sum of the cardinalities; or

(1.2.4b)

The generalization of this result to an arbitrary union of n sets is called the inclusion–exclusion principle, given by

(1.2.5a)

If the sets {Ai} are mutually exclusive, that is, Ai∩Aj = Ø for i≠j, then we have

(1.2.5b)

This equation is illustrated in the Venn diagram for n = 3 in Fig. 1.2.3, where if equals, then we have added twice the cardinalites of and . However, if we subtract once , , and and write

then we have subtracted thrice instead of twice. Hence, adding we get...

Produkt-Details

• Erscheint lt. Verlag: 15.7.2015
• Co-Autor: Kavitha Chandra
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Angew. Wahrscheinlichkeitsrechn. u. Statistik / Modelle • Applied Probability & Statistics - Models • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Engineering statistics • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen • Statistics • Statistik • Statistik in den Ingenieurwissenschaften
• ISBN-10: 1-119-01190-6 / 1119011906
• ISBN-13: 978-1-119-01190-3 / 9781119011903