Fluent Essays

SLOVED PROBLEMS 2.1 Consider the experiment of throwing a fair die. Let X be the r.v. which assigns 1 if the number that appears is even and 0 if the number that appears is odd. (a) What is the range of X? (b) Find P(X = 1) and P(X = 0). 2.2 Consider the experiment of tossing a coin three times (Prob. 1.1). Let X be the r.v. giving the number of heads obtained. We assume that the tosses are independent and the probability of a head is p. (a) What is the range of X? (b) Find the probabilities P(X = 0), P(X = 1), P(X = 2), and P(X = 3). The sample space S on which X is defined consists of eight sample points (Prob. 1.1): 2.3 An information source generates symbols at random from a four-letter alphabet {a, b, c, d} with probabilities , and . A coding scheme encodes these symbols into binary codes as follows: Let X be the r.v. denoting the length of the code—that is, the number of binary symbols (bits). (a) What is the range of X? (b) Assuming that the generations of symbols are independent, find the probabilities P(X = 1), P(X = 2), P(X = 3), and P(X > 3). TEXT BOOK PG NO : 111 2.5. Verify Eq. (2.6). Let x1 < x2 . Then (X ≤ x1 ) is a subset of (X ≤ x2 ); that is, (X ≤ x1 ) ⊂ (X ≤ x2 ). Then, by Eq. (1.41), we have. 2.6 Verify (a) Eq. (2.10); (b) Eq. (2.11); (c) Eq. (2.12). (a) Since (X ≤ b) = (X ≤ a) ∪ (a < X ≤ b) and (X ≤ a) ∩ (a < X ≤ b) = ∅, we have. 2.7 Textbook pg no : 113 2.8. Let X be the r.v. defined in Prob. 2.3. (a) Sketch the cdf FX (x) of X and specify the type of X. (b) Find (i) P(X ≤ 1), (ii) P(1 < X ≤ 2), (iii) P(X > 1), and (iv) P(1 ≤ X ≤ 2). (a) From the result of Prob. 2.3 and Eq. (2.18), we have which is sketched in Fig. 2-15. The r.v. X is a discrete r.v. Textbook pg no 114 2.10 Consider the function given by (a) Sketch F(x) and show that F(x) has the properties of a cdf discussed in Sec. 2.3B. (b) If X is the r.v. whose cdf is given by F(x), find (i) , (ii) , (iii) P(X = 0), and (iv) . (c) Specify the type of X. Textbook pg no :117 2.11 Textbook pg 118 2.12 2.13 2.15 Textbook 122 2.19 Textbook 125 2.20 Texbook 126 2.21 textbook no 126 2.22 Textbook pg no: 127 2.53 Textbook pg no : 151 2.54 Textbook pg no : 152 Practice Problems from Schaum Series (Functions of One Random Variable): Solved problems 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.8, 4.9, 4.14 problems 4.1 TextBook pg no : 245 problems 4.2 TextBook pg no : 246 problems 4.3 TextBook pg no : 247 problems 4.4 TextBook pg no : 249 problems 4.5 TextBook pg no : 250 problems 4.6 TextBook pg no : 251 problems 4.8 TextBook pg no : 252 problems 4.9 TextBook pg no : 253 problems 4.14 TextBook pg no : 254 Practice Problems from Schaum Series (Mean, Variance, Moments): Solved Problems: 2.27, 2.28, 2.32, 2.33, 2.35, 4.42, 4.43 Solved Problems: 2.27 pg no : 132 Solved Problems: 2.28 pg no : 134 Solved Problems: 2.32 pg no : 138 Solved Problems: 2.33 pg no : 138 Solved Problems: 2.35 pg no : 139 Solved Problems: 4.42 pg no : 280 Solved Problems: 4.43 pg no : 280 HWEI P. HSU received his BS from National Taiwan University and his MS and PhD from Case Institute of Technology. He has published several books which include Schaum’s Outline of Analog and Digital Communications and Schaum’s Outline of Signals and Systems. Copyright © 2020, 2013, 2009, 1998 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. ISBN: 978-1-26-045382-9 MHID: 1-26-045382-0 The material in this eBook also appears in the print version of this title: ISBN: 978-1-26-045381-2, MHID: 1-26-045381-2. eBook conversion by codeMantra Version 1.0 All trademarks are trademarks of their respective owners. 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Under no circumstances shall McGraw-Hill Education and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. http://www.schaums.com/ Preface to The Second Edition The purpose of this book, like its previous edition, is to provide an introduction to the principles of probability, random variables, and random processes and their applications. The book is designed for students in various disciplines of engineering, science, mathematics, and management. The background required to study the book is one year of calculus, elementary differential equations, matrix analysis, and some signal and system theory, including Fourier transforms. The book can be used as a self-contained textbook or for self-study. Each topic is introduced in a chapter with numerous solved problems. The solved problems constitute an integral part of the text. This new edition includes and expands the contents of the first edition. In addition to refinement through the text, two new sections on probability- generating functions and martingales have been added and a new chapter on information theory has been added. I wish to thank my granddaughter Elysia Ann Krebs for helping me in the preparation of this revision. I also wish to express my appreciation to the editorial staff of the McGraw-Hill Schaum’s Series for their care, cooperation, and attention devoted to the preparation of the book. HWEI P. HSU Shannondell at Valley Forge, Audubon, Pennsylvania Preface to The First Edition The purpose of this book is to provide an introduction to the principles of probability, random variables, and random processes and their applications. The book is designed for students in various disciplines of engineering, science, mathematics, and management. It may be used as a textbook and/or a supplement to all current comparable texts. It should also be useful to those interested in the field of self-study. The book combines the advantages of both the textbook and the so-called review book. It provides the textual explanations of the textbook, and in the direct way characteristic of the review book, it gives hundreds of completely solved problems that use essential theory and techniques. Moreover, the solved problems are an integral part of the text. The background required to study the book is one year of calculus, elementary differential equations, matrix analysis, and some signal and system theory, including Fourier transforms. I wish to thank Dr. Gordon Silverman for his invaluable suggestions and critical review of the manuscript. I also wish to express my appreciation to the editorial staff of the McGraw-Hill Schaum Series for their care, cooperation, and attention devoted to the preparation of the book. Finally, I thank my wife, Daisy, for her patience and encouragement. HWEI P. HSU Montville, New Jersey Contents CHAPTER 1 Probability 1.1 Introduction 1.2 Sample Space and Events 1.3 Algebra of Sets 1.4 Probability Space 1.5 Equally Likely Events 1.6 Conditional Probability 1.7 Total Probability 1.8 Independent Events Solved Problems CHAPTER 2 Random Variables 2.1 Introduction 2.2 Random Variables 2.3 Distribution Functions 2.4 Discrete Random Variables and Probability Mass Functions 2.5 Continuous Random Variables and Probability Density Functions 2.6 Mean and Variance 2.7 Some Special Distributions 2.8 Conditional Distributions Solved Problems CHAPTER 3 Multiple Random Variables 3.1 Introduction 3.2 Bivariate Random Variables 3.3 Joint Distribution Functions 3.4 Discrete Random Variables—Joint Probability Mass Functions 3.5 Continuous Random Variables—Joint Probability Density Functions 3.6 Conditional Distributions 3.7 Covariance and Correlation Coefficient 3.8 Conditional Means and Conditional Variances 3.9 N-Variate Random Variables 3.10 Special Distributions Solved Problems CHAPTER 4 Functions of Random Variables, Expectation, Limit Theorems 4.1 Introduction 4.2 Functions of One Random Variable 4.3 Functions of Two Random Variables 4.4 Functions of n Random Variables 4.5 Expectation 4.6 Probability Generating Functions 4.7 Moment Generating Functions 4.8 Characteristic Functions 4.9 The Laws of Large Numbers and the Central Limit Theorem Solved Problems CHAPTER 5 Random Processes 5.1 Introduction 5.2 Random Processes 5.3 Characterization of Random Processes 5.4 Classification of Random Processes 5.5 Discrete-Parameter Markov Chains 5.6 Poisson Processes 5.7 Wiener Processes 5.8 Martingales Solved Problems CHAPTER 6 Analysis and Processing of Random Processes 6.1 Introduction 6.2 Continuity, Differentiation, Integration 6.3 Power Spectral Densities 6.4 White Noise 6.5 Response of Linear Systems to Random Inputs 6.6 Fourier Series and Karhunen-Loéve Expansions 6.7 Fourier Transform of Random Processes Solved Problems CHAPTER 7 Estimation Theory 7.1 Introduction 7.2 Parameter Estimation 7.3 Properties of Point Estimators 7.4 Maximum-Likelihood Estimation 7.5 Bayes’ Estimation 7.6 Mean Square Estimation 7.7 Linear Mean Square Estimation Solved Problems CHAPTER 8 Decision Theory 8.1 Introduction 8.2 Hypothesis Testing 8.3 Decision Tests Solved Problems CHAPTER 9 Queueing Theory 9.1 Introduction 9.2 Queueing Systems 9.3 Birth-Death Process 9.4 The M/M/1 Queueing System 9.5 The M/M/s Queueing System 9.6 The M/M/1/K Queueing System 9.7 The M/M/s/K Queueing System Solved Problems CHAPTER 10 Information Theory 10.1 Introduction 10.2 Measure of Information 10.3 Discrete Memoryless Channels 10.4 Mutual Information 10.5 Channel Capacity 10.6 Continuous Channel 10.7 Additive White Gaussian Noise Channel 10.8 Source Coding 10.9 Entropy Coding Solved Problems APPENDIX A Normal Distribution APPENDIX B Fourier Transform B.1 Continuous-Time Fourier Transform B.2 Discrete-Time Fourier Transform INDEX *The laptop icon next to an exercise indicates that the exercise is also available as a video with step- by-step instructions. These videos are available on the Schaums.com website by following the instructions on the inside front cover. CHAPTER 1 Probability 1.1 Introduction The study of probability stems from the analysis of certain games of chance, and it has found applications in most branches of science and engineering. In this chapter the basic concepts of probability theory are presented. 1.2 Sample Space and Events A. Random Experiments: In the study of probability, any process of observation is referred to as an experiment. The results of an observation are called the outcomes of the experiment. An experiment is called a random experiment if its outcome cannot be predicted. Typical examples of a random experiment are the roll of a die, the toss of a coin, drawing a card from a deck, or selecting a message signal for transmission from several messages. B. Sample Space: The set of all possible outcomes of a random experiment is called the sample space (or universal set), and it is denoted by S. An element in S is called a sample point. Each outcome of a random experiment corresponds to a sample point. EXAMPLE 1.1 Find the sample space for the experiment of tossing a coin (a) once and (b) twice. (a) There are two possible outcomes, heads or tails. Thus: where H and T represent head and tail, respectively. (b) There are four possible outcomes. They are pairs of heads and tails. Thus: EXAMPLE 1.2 Find the sample space for the experiment of tossing a coin repeatedly and of counting the number of tosses required until the first head appears. Clearly all possible outcomes for this experiment are the terms of the sequence 1, 2, 3, … Thus: Note that there are an infinite number of outcomes. EXAMPLE 1.3 Find the sample space for the experiment of measuring (in hours) the lifetime of a transistor. Clearly all possible outcomes are all nonnegative real numbers. That is, where τ represents the life of a transistor in hours. Note that any particular experiment can often have many different sample spaces depending on the observation of interest (Probs. 1.1 and 1.2). A sample space S is said to be discrete if it consists of a finite number of sample points (as in Example 1.1) or countably infinite sample points (as in Example 1.2). A set is called countable if its elements can be placed in a one-to-one correspondence with the positive integers. A sample space S is said to be continuous if the sample points constitute a continuum (as in Example 1.3). C. Events: Since we have identified a sample space S as the set of all possible outcomes of a random experiment, we will review some set notations in the following. If ζ is an element of S (or belongs to S), then we write If S is not an element of S (or does not belong to S), then we write A set A is called a subset of B, denoted by if every element of A is also an element of B. Any subset of the sample space S is called an event. A sample point of S is often referred to as an elementary event. Note that the sample space S is the subset of itself: that is, S ⊂ S. Since S is the set of all possible outcomes, it is often called the certain event. EXAMPLE 1.4 Consider the experiment of Example 1.2. Let A be the event that the number of tosses required until the first head appears is even. Let B be the event that the number of tosses required until the first head appears is odd. Let C be the event that the number of tosses required until the first head appears is less than 5. Express events A, B, and C. 1.3 Algebra of Sets A. Set Operations: 1. Equality: Two sets A and B are equal, denoted A = B, if and only if A ⊂ B and B ⊂ A. 2. Complementation: Suppose A ⊂ S. The complement of set A, denoted , is the set containing all elements in S but not in A. 3. Union: The union of sets A and B, denoted A ∪ B, is the set containing all elements in either A or B or both. 4. Intersection: The intersection of sets A and B, denoted A ∩ B, is the set containing all elements in both A and B. 5. Difference: The difference of sets A and B, denoted A\ B, is the set containing all elements in A but not in B. Note that . 6. Symmetrical Difference: The symmetrical difference of sets A and B, denoted A Δ B, is the set of all elements that are in A or B but not in both. Note that . 7. Null Set: The set containing no element is called the null set, denoted Ø. Note that 8. Disjoint Sets: Two sets A and B are called disjoint or mutually exclusive if they contain no common element, that is, if A ∩ B = Ø. The definitions of the union and intersection of two sets can be extended to any finite number of sets as follows: Note that these definitions can be extended to an infinite number of sets: In our definition of event, we state that every subset of S is an event, including S and the null set Ø Then If A and B are events in S, then Similarly, if A1, A2, …, An are a sequence of events in S, then 9. Partition of S : If Ai ∩ Aj = Ø for i ≠ j and , then the collection {Ai; 1 ≤ i ≤ k} is said to form a partition of S. 10. Size of Set: When sets are countable, the size (or cardinality) of set A, denoted |A|, is the number of elements contained in A. When sets have a finite number of elements, it is easy to see that size has the following properties: Note that the property (iv) can be easily seen if A and B are subsets of a line with length |A| and |B|, respectively. 11. Product of Sets: The product (or Cartesian product) of sets A and B, denoted by A × B, is the set of ordered pairs of elements from A and B. Note that A × B ≠ B × A, and |C| = |A × B| = |A| × |B|. EXAMPLE 1.5 Let A = {a1, a2, a3} and B = {b1, b2}. Then B. Venn Diagram: A graphical representation that is very useful for illustrating set operation is the Venn diagram. For instance, in the three Venn diagrams shown in Fig. 1-1, the shaded areas represent, respectively, the events A ∪ B, A ∩ B, and . The Venn diagram in Fig. 1-2(a) indicates that B ⊂ A, and the event A ∩ = A\B is shown as the shaded area. In Fig. 1-2(b), the shaded area represents the event A Δ B. Fig. 1-1 Fig. 1-2 C. Identities: By the above set definitions or reference to Fig. 1-1, we obtain the following identities: The union and intersection operations also satisfy the following laws: Commutative Laws: Associative Laws: Distributive Laws: De Morgan’s Laws: These relations are verified by showing that any element that is contained in the set on the left side of he equality sign is also contained in the set on the right side, and vice versa. One way of showing this is by means of a Venn diagram (Prob. 1.14). The distributive laws can be extended as follows: Similarly, De Morgan’s laws also can be extended as follows (Prob. 1.21): 1.4 Probability Space A. Event Space: We have defined that events are subsets of the sample space S. In order to be precise, we say that a subset A of S can be an event if it belongs to a collection F of subsets of S, satisfying the following conditions: The collection F is called an event space. In mathematical literature, event space is known as sigma field (σ-field) or σ-algebra. Using the above conditions, we can show that if A and B are in F, then so are EXAMPLE 1.6 Consider the experiment of tossing a coin once in Example 1.1. We have S = {H, T}. The set {S, Ø}, {S, Ø, H, T} are event spaces, but {S, Ø, H} is not an event space, since = T is not in the set. B. Probability Space: An assignment of real numbers to the events defined in an event space F is known as the probability measure P. Consider a random experiment with a sample space S, and let A be a particular event defined in F. The probability of the event A is denoted by P(A). Thus, the probability measure is a function defined over F. The triplet (S, F, P) is known as the probability space. C. Probability Measure a. Classical Definition: Consider an experiment with equally likely finite outcomes. Then the classical definition of probability of event A, denoted P(A), is defined by If A and B are disjoint, i.e., A ∩ B = Ø, then, |A ∪ B| = |A| + |B|. Hence, in this case We also have EXAMPLE 1.7 Consider an experiment of rolling a die. The outcome is Define: A: the event that outcome is even, i.e., A = {2, 4, 6} B: the event that outcome is odd, i.e., B = {1, 3, 5} C: the event that outcome is prime, i.e., C = {1, 2, 3, 5} Then Note that in the classical definition, P(A) is determined a priori without actual experimentation and the definition can be applied only to a limited class of problems such as only if the outcomes are finite and equally likely or equally probable. b. Relative Frequency Definition: Suppose that the random experiment is repeated n times. If event A occurs n(A) times, then the probability of event A, denoted P(A), is defined as where n(A)/n is called the relative frequency of event A. Note that this limit may not exist, and in addition, there are many situations in which the concepts of repeatability may not be valid. It is clear that for any event A, the relative frequency of A will have the following properties: 1. 0 ≤ n(A)/n ≤ 1, where n(A)/n = 0 if A occurs in none of the n repeated trials and n(A)/n = 1 if A occurs in all of the n repeated trials. 2. If A and B are mutually exclusive events, then and c. Axiomatic Definition: Consider a probability space (S, F, P). Let A be an event in F. Then in the axiomatic definition, the probability P(A) of the event A is a real number assigned to A which satisfies the following three axioms: If the sample space S is not finite, then axiom 3 must be modified as follows: Axiom 3′: If A1, A2, … is an infinite sequence of mutually exclusive events in S (Ai ∩ Aj = Ø for i ≠ j), then These axioms satisfy our intuitive notion of probability measure obtained from the notion of relative frequency. d. Elementary Properties of Probability: By using the above axioms, the following useful properties of probability can be obtained: where the sum of the second term is over all distinct pairs of events, that of the third term is over all distinct triples of events, and so forth. 8. If A1, A2, …, An is a finite sequence of mutually exclusive events in S (Ai ∩ Aj = Ø for i ≠ j), then and a similar equality holds for any subcollection of the events. Note that property 4 can be easily derived from axiom 2 and property 3. Since A ⊂ S, we have Thus, combining with axiom 1, we obtain Property 5 implies that since P(A ∩ B) ≥ 0 by axiom 1. Property 6 implies that since A ∩ B = B if B ⊂ A. 1.5 Equally Likely Events A. Finite Sample Space: Consider a finite sample space S with n finite elements where ζi’s are elementary events. Let P(ζi) = pi. Then B. Equally Likely Events: When all elementary events ζi(i = 1, 2, …, n) are equally likely, that is, then from Eq. (1.51), we have and where n(A) is the number of outcomes belonging to event A and n is the number of sample points in S. [See classical definition (1.29).] 1.6 Conditional Probability A. Definition: The conditional probability of an event A given event B, denoted by P(A | B), is defined as where P(A ∩ B) is the joint probability of A and B. Similarly, is the conditional probability of an event B given event A. From Eqs. (1.55) and (1.56), we have Equation (1.57) is often quite useful in computing the joint probability of events. B. Bayes’ Rule: From Eq. (1.57) we can obtain the following Bayes’ rule: 1.7 Total Probability The events A1, A2, …, An are called mutually exclusive and exhaustive if Let B be any event in S. Then which is known as the total probability of event B (Prob. 1.57). Let A = Ai in Eq. (1.58); then, using Eq. (1.60), we obtain Note that the terms on the right-hand side are all conditioned on events Ai, while the term on the left is conditioned on B. Equation (1.61) is sometimes referred to as Bayes’ theorem. 1.8 Independent Events Two events A and B are said to be (statistically) independent if and only if It follows immediately that if A and B are independent, then by Eqs. (1.55) and (1.56), If two events A and B are independent, then it can be shown that A and are also independent; that is (Prob. 1.63), Then Thus, if A is independent of B, then the probability of A‘s occurrence is unchanged by information as to whether or not B has occurred. Three events A, B, C are said to be independent if and only if We may also extend the definition of independence to more than three events. The events A1, A2, …, An are independent if and only if for every subset {Ai1, Ai2, … Aik} (2 ≤ k ≤ n) of these events, Finally, we define an infinite set of events to be independent if and only if every finite subset of these events is independent. To distinguish between the mutual exclusiveness (or disjointness) and independence of a collection of events, we summarize as follows: 1. {Ai, i = 1, 2, …, n} is a sequence of mutually exclusive events, then 2. If {Ai, i = 1, 2, …, n} is a sequence of independent events, then and a similar equality holds for any subcollection of the events. SOLVED PROBLEMS Sample Space and Events 1.1. Consider a random experiment of tossing a coin three times. (a) Find the sample space S1 if we wish to observe the exact sequences of heads and tails obtained. (b) Find the sample space S2 if we wish to observe the number of heads in the three tosses. (a) The sampling space S1 is given by where, for example, HTH indicates a head on the first and third throws and a tail on the second throw. There are eight sample points in S1. (b) The sampling space S2 is given by where, for example, the outcome 2 indicates that two heads were obtained in the three tosses. The sample space S2 contains four sample points. 1.2. Consider an experiment of drawing two cards at random from a bag containing four cards marked with the integers 1 through 4. (a) Find the sample space S1 of the experiment if the first card is replaced before the second is drawn. (b) Find the sample space S2 of the experiment if the first card is not replaced. (a) The sample space S1 contains 16 ordered pairs (i, j), 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, where the first number indicates the first number drawn. Thus, (b) The sample space S2 contains 12 ordered pairs (i, j), i ≠ j, 1 ≤ i ≤ 4, 1 ≤ j ≤ 4, where the first number indicates the first number drawn. Thus, 1.3. An experiment consists of rolling a die until a 6 is obtained. (a) Find the sample space S1 if we are interested in all possibilities. (b) Find the sample space S2 if we are interested in the number of throws needed to get a 6. (a) The sample space S1 would be where the first line indicates that a 6 is obtained in one throw, the second line indicates that a 6 is obtained in two throws, and so forth. (b) In this case, the sample space S2 is where i is an integer representing the number of throws needed to get a 6. 1.4. Find the sample space for the experiment consisting of measurement of the voltage output v from a transducer, the maximum and minimum of which are + 5 and − 5 volts, respectively. A suitable sample space for this experiment would be 1.5. An experiment consists of tossing two dice. (a) Find the sample space S. (b) Find the event A that the sum of the dots on the dice equals 7. (c) Find the event B that the sum of the dots on the dice is greater than 10. (d) Find the event C that the sum of the dots on the dice is greater than 12. (a) For this experiment, the sample space S consists of 36 points (Fig. 1-3): Fig. 1-3 where i represents the number of dots appearing on one die and j represents the number of dots appearing on the other die. (b) The event A consists of 6 points (see Fig. 1-3): (c) The event B consists of 3 points (see Fig. 1-3): (d) The event C is an impossible event, that is, C = Ø. 1.6. An automobile dealer offers vehicles with the following options: (a) With or without automatic transmission (b) With or without air-conditioning (c) With one of two choices of a stereo system (d) With one of three exterior colors If the sample space consists of the set of all possible vehicle types, what is the number of outcomes in the sample space? The tree diagram for the different types of vehicles is shown in Fig. 1-4. From Fig. 1-4 we see that the number of sample points in S is 2 × 2 × 2 × 3 = 24. Fig. 1-4 1.7. State every possible event in the sample space S = {a, b, c, d}. There are 24= 16 possible events in S. They are Ø; {a}, {b}, {c}, {d}; {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}; {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}; S = {a, b, c, d}. 1.8. How many events are there in a sample space S with n elementary events? Let S = {s1, s2, …, sn}. Let Ω be the family of all subsets of S. (Ω is sometimes referred to as the power set of S.) Let Si be the set consisting of two statements, that is, Then Ω can be represented as the Cartesian product Since each subset of S can be uniquely characterized by an element in the above Cartesian product, we obtain the number of elements in Ω by where n(Si) = number of elements in Si = 2. An alternative way of finding n(Ω) is by the following summation: The last sum is an expansion of (1+1)n = 2n. Algebra of Sets 1.9. Consider the experiment of Example 1.2. We define the events where k is the number of tosses required until the first H (head) appears. Determine the events , , , A ∪ B, B ∪ C, A ∩ B, A ∩ C, B ∩ C, and ∩ B. 1.10. Consider the experiment of Example 1.7 of rolling a die. Express From Example 1.7, we have S = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {1, 3, 5}, and C = {1, 2, 3, 5}. Then 1.11. The sample space of an experiment is the real line express as (a) Consider the events Determine the events (b) Consider the events Determine the events (a) It is clear that Noting that the Ai’s are mutually exclusive, we have (b) Noting that B1 ⊃ B2 ⊃ … ⊃ Bi ⊃ …, we have 1.12. Consider the switching networks shown in Fig. 1-5. Let A1, A2, and A3 denote the events that the switches s1, s2, and s3 are closed, respectively. Let Aab denote the event that there is a closed path between terminals a and b. Express Aab in terms of A1, A2, and A3 for each of the networks shown. Fig. 1-5 (a) From Fig. 1-5(a), we see that there is a closed path between a and b only if all switches s1, s2, and s3 are closed. Thus, (b) From Fig. 1-5(b), we see that there is a closed path between a and b if at least one switch is closed. Thus, (c) From Fig. 1-5(c), we see that there is a closed path between a and b if s1 and either s2 or s3 are closed. Thus, Using the distributive law (1.18), we have which indicates that there is a closed path between a and b if s1 and s2 or s1 and s3 are closed. (d) From Fig. 1-5(d), we see that there is a closed path between a and b if either s1 and s2 are closed or s3 is closed. Thus 1.13. Verify the distributive law (1.18). Let s ∈ [A ∩ (B ∪ C)]. Then s ∈ A and s ∈ (B ∪ C). This means either that s ∈ A and s ∈ B or that s ∈ A and s ∈ C; that is, s …