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Other Books in Schaum’s Easy Outline Series Include: Schaum’s Easy Outline: College Mathematics Schaum’s Easy Outline: College Algebra Schaum’s Easy Outline: Calculus Schaum’s Easy Outline: Elementary Algebra Schaum’s Easy Outline: Mathematical Handbook of Formulas and Tables Schaum’s Easy Outline: Geometry Schaum’s Easy Outline: Precalculus Schaum’s Easy Outline: Trigonometry Schaum’s Easy Outline: Probability and Statistics Schaum’s Easy Outline: Statistics Schaum’s Easy Outline: Principles of Accounting Schaum’s Easy Outline: Biology Schaum’s Easy Outline: College Chemistry Schaum’s Easy Outline: Genetics Schaum’s Easy Outline: Human Anatomy and Physiology Schaum’s Easy Outline: Organic Chemistry Schaum’s Easy Outline: Physics Schaum’s Easy Outline: Programming with C++ Schaum’s Easy Outline: Programming with Java Schaum’s Easy Outline: French Schaum’s Easy Outline: German Schaum’s Easy Outline: Spanish Schaum’s Easy Outline: Writing and Grammar
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SCHAUM’S Easy OUTLINES PROBABILITY AND STATISTICS BA S E D O N SC H A U M’S Outline of Probability and Statistics BY MURRAY R. SPIEGEL, JOHN SCHILLER, AND R. ALU SRINIVASAN AB R I D G M E N T ED I T O R MI K E LEVA N S C H A U M ’ S O U T L I N E S E R I E S MCG R AW- H I L L New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
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McGraw-Hillabc Copyright © 2001 by The McGraw-Hill Companies,Inc. All rights reserved. Manufactured in the United States of America. 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 data- base or retrieval system, without the prior written permission of the publisher. 0-07-139838-4 The material in this eBook also appears in the print version of this title:0-07-138341-7 All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales pro- motions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS”. McGRAW-HILL AND ITS LICENSORS MAKE NO GUAR- ANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMA- TION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the func- tions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inac- curacy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill 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 lia- bility shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/0071398384
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For more information about this book, click here. Contents Chapter 1 Basic Probability 1 Chapter 2 Descriptive Statistics 14 Chapter 3 Discrete Random Variables 23 Chapter 4 Continuous Random Variables 34 Chapter 5 Examples of Random Variables 42 Chapter 6 Sampling Theory 58 Chapter 7 Estimation Theory 75 Chapter 8 Test of Hypothesis and Significance 85 Chapter 9 Curve Fitting, Regression, and Correlation 99 Chapter 10 Other Probability Distributions 117 Appendix A Mathematical Topics 132 Appendix B Areas under the Standard Normal Curve from 0 to z 136 Appendix C Student’s t distribution 138 Appendix D Chi-Square Distribution 140 Appendix E95th and 99th Percentile Values for the F Distribution 142 Appendix F Values of e−λ 146 Appendix G Random Numbers 148 Index 149 v Copyright 2001 by the McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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Chapter 1 BASIC PROBABILITY IN THIS CHAPTER: ✔ Random Experiments ✔ Sample Spaces ✔ Events ✔ The Concept of Probability ✔ The Axioms of Probability ✔ Some Important Theorems on Probability ✔ Assignment of Probabilities ✔ Conditional Probability ✔ Theorem on Conditional Probability ✔ Independent Events ✔ Bayes’ Theorem or Rule ✔ Combinatorial Analysis ✔ Fundamental Principle of Counting ✔ Permutations ✔ Combinations 1 Copyright 2001 by the McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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2 PROBABILITY AND STATISTICS ✔ Binomial Coefficients ✔ Stirling’s Approximation to n! Random Experiments We are all familiar with the importance of experi- ments in science and engineering. Experimentation is useful to us because we can assume that if we perform certain experiments under very nearly identical conditions, we will arrive at results that are essentially the same. In these circumstances, we are able to control the value of the variables that affect the outcome of the experiment. However, in some experiments, we are not able to ascertain or con- trol the value of certain variables so that the results will vary from one performance of the experiment to the next, even though most of the con- ditions are the same. These experiments are described as random. Here is an example: Example 1.1. If we toss a die, the result of the experiment is that it will come up with one of the numbers in the set {1, 2, 3, 4, 5, 6}. Sample Spaces A set S that consists of all possible outcomes of a random experiment is called a sample space, and each outcome is called a sample point. Often there will be more than one sample space that can describe outcomes of an experiment, but there is usually only one that will provide the most information. Example 1.2. If we toss a die, then one sample space is given by {1, 2, 3, 4, 5, 6} while another is {even, odd}. It is clear, however, that the latter would not be adequate to determine, for example, whether an outcome is divisible by 3. If is often useful to portray a sample space graphically. In such cases, it is desirable to use numbers in place of letters whenever possible.
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CHAPTER 1: Basic Probability 3 If a sample space has a finite number of points, it is called a finite sample space. If it has as many points as there are natural numbers 1, 2, 3, …. , it is called a countably infinite sample space. If it has as many points as there are in some interval on the x axis, such as 0 ≤ x ≤ 1, it is called a noncountably infinite sample space. A sample space that is finite or countably finite is often called a discrete sample space, while one that is noncountably infinite is called a nondiscrete sample space. Example 1.3. The sample space resulting from tossing a die yields a discrete sample space. However, picking any number, not just inte- gers, from 1 to 10, yields a nondiscrete sample space. Events An event is a subset A of the sample space S, i.e., it is a set of possible outcomes. If the outcome of an experiment is an element of A, we say that the event A has occurred. An event consisting of a single point of S is called a simple or elementary event. As particular events, we have S itself, which is the sure or certain event since an element of S must occur, and the empty set ∅, which is called the impossible event because an element of ∅ cannot occur. By using set operations on events in S, we can obtain other events in S. For example, if A and B are events, then 1. A ∪ B is the event “either A or B or both.” A ∪ B is called the union of A and B. 2. A ∩ B is the event “both A and B.” A ∩ B is called the inter- section of A and B. 3. A′ is the event “not A.” A′ is called the complement of A. 4. A – B = A ∩ B′ is the event “A but not B.” In particular, A′ = S – A. If the sets corresponding to events A and B are disjoint, i.e., A ∩ B = ∅, we often say that the events are mutually exclusive. This means that they cannot both occur. We say that a collection of events A , A , … , 1 2 A is mutually exclusive if every pair in the collection is mutually exclu- n sive.
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4 PROBABILITY AND STATISTICS The Concept of Probability In any random experiment there is always uncertainty as to whether a particular event will or will not occur. As a measure of the chance, or probability, with which we can expect the event to occur, it is conve- nient to assign a number between 0 and 1. If we are sure or certain that an event will occur, we say that its probability is 100% or 1. If we are sure that the event will not occur, we say that its probability is zero. If, for example, the probability is ¹⁄ , we would say that there is a 25% chance it will occur and a 75% chance that it will not occur. Equivalently, we can say that the odds against occurrence are 75% to 25%, or 3 to 1. There are two important procedures by means of which we can esti- mate the probability of an event. 1. CLASSICAL APPROACH: If an event can occur in h different ways out of a total of n possible ways, all of which are equally likely, then the probability of the event is h/n. 2. FREQUENCY APPROACH: If after n repetitions of an experiment, where n is very large, an event is observed to occur in h of these, then the probability of the event is h/n. This is also called the empirical probability of the event. Both the classical and frequency approaches have serious drawbacks, the first because the words “equally likely” are vague and the second because the “large number” involved is vague. Because of these difficulties, mathematicians have been led to an axiomatic approach to probability. The Axioms of Probability Suppose we have a sample space S. If S is discrete, all subsets corre- spond to events and conversely; if S is nondiscrete, only special subsets (called measurable) correspond to events. To each event A in the class C of events, we associate a real number P(A). The P is called a proba- bility function, and P(A) the probability of the event, if the following axioms are satisfied.
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CHAPTER 1: Basic Probability 5 Axiom 1. For every event A in class C, P(A) ≥ 0 Axiom 2. For the sure or certain event S in the class C, P(S) = 1 Axiom 3. For any number of mutually exclusive events A , A , …, 1 2 in the class C, P(A ∪ A ∪ … ) = P(A ) + P(A ) + … 1 2 1 2 In particular, for two mutually exclusive events A and A , 1 2 P(A ∪ A ) = P(A ) + P(A ) 1 2 1 2 Some Important Theorems on Probability From the above axioms we can now prove various theorems on proba- bility that are important in further work. Theorem 1-1: If A ⊂ A , then (1) 1 2 P(A ) ≤ P(A ) and P(A − A ) = P(A ) − P(A ) 1 2 2 1 1 2 Theorem 1-2: For every event A, (2) 0 ≤ P(A) ≤ 1, i.e., a probability between 0 and 1. Theorem 1-3: For ∅, the empty set, (3) P(∅) = 0 i.e., the impossible event has probability zero. Theorem 1-4: If A′ is the complement of A, then (4) P( A′) = 1 – P(A) Theorem 1-5: If A = A ∪ A ∪ … ∪ A , where A A A are 1 2 n 1, 2, … , n mutually exclusive events, then P(A) = P(A ) + P(A ) + … + P(A ) (5) 1 2 n
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6 PROBABILITY AND STATISTICS Theorem 1-6: If A and B are any two events, then (6) P(A ∪ B) = P(A) + P(B) – P(A ∩ B) More generally, if A , A , A are any three events, 1 2 3 then P(A ∪ A ∪ A ) = P(A ) + P(A ) + P(A ) – 1 2 3 1 2 3 P(A ∩ A ) – P(A ∩ A ) – P(A ∩ A ) + 1 2 2 3 3 1 P(A ∩ A ∩ A ). 1 2 3 Generalizations to n events can also be made. Theorem 1-7: For any events A and B, (7) P(A) = P(A ∩ B) + P(A ∩ B′) Assignment of Probabilities If a sample space S consists of a finite number of outcomes a , a , … , 1 2 a , then by Theorem 1-5, n P(A ) + P(A ) + … + P(A ) = 1 (8) 1 2 n where A , A , … , A are elementary events given by A = {a }. 1 2 n i i It follows that we can arbitrarily choose any nonnegative numbers for the probabilities of these simple events as long as the previous equa- tion is satisfied. In particular, if we assume equal probabilities for all simple events, then 1 P A( k ) = , k = 1, 2, … , n (9) n And if A is any event made up of h such simple events, we have h P A( ) = (10) n This is equivalent to the classical approach to probability. We could of course use other procedures for assigning probabilities, such as fre- quency approach.
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CHAPTER 1: Basic Probability 7 Assigning probabilities provides a mathematical model, the success of which must be tested by experiment in much the same manner that the theories in physics or others sciences must be tested by experiment. Remember The probability for any event must be between 0 and 1. Conditional Probability Let A and B be two events such that P(A) > 0. Denote P(B | A) the prob- ability of B given that A has occurred. Since A is known to have occurred, it becomes the new sample space replacing the original S. From this we are led to the definition P A( ∩ B) P B( | A) ≡ (11) P A( ) or P A( ∩ B) ≡ P A P B( ) ( | A) (12) In words, this is saying that the probability that both A and B occur is equal to the probability that A occurs times the probability that B occurs given that A has occurred. We call P(B | A) the conditional prob- ability of B given A, i.e., the probability that B will occur given that A has occurred. It is easy to show that conditional probability satisfies the axioms of probability previously discussed. Theorem on Conditional Probability Theorem 1-8: For any three events A , A , A , we have 1 2 3 P A( 1 ∩ A2 ∩ A3) = P A( 1)P A( 2 | A1)P A( 3 | A1 ∩ A2 ) (13)
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8 PROBABILITY AND STATISTICS In words, the probability that A and A and A all occur is equal 1 2 3 to the probability that A occurs times the probability that A occurs 1 2 given that A has occurred times the probability that A occurs given 1 3 that both A and A have occurred. The result is easily generalized to n 1 2 events. Theorem 1-9: If an event A must result in one of the mutually exclusive events A , A , … , A , then P(A) 1 2 n = P(A )P(A | A ) + P(A )P(A | A ) +... 1 1 2 2 + P(A )P(A | A ) (14) n n Independent Events If P(B | A) = P(B), i.e., the probability of B occurring is not affected by the occurrence or nonoccurrence of A, then we say that A and B are independent events. This is equivalent to P A( ∩ B) = P A P B( ) ( ) (15) Notice also that if this equation holds, then A and B are indepen- dent. We say that three events A1, A2, A3 are independent if they are pairwise independent. P(A ∩ A ) = P(A )P(A ) j ≠ k where j,k = 1,2,3 (16) j k j k and P A( 1 ∩ A2 ∩ A3) = P A( 1)P A( 2 )P A( 3) (17) Both of these properties must hold in order for the events to be independent. Independence of more than three events is easily defined.
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CHAPTER 1: Basic Probability 9 Note! In order to use this multiplication rule, all of your events must be independent. Bayes’ Theorem or Rule Suppose that A , A , … , A are mutually exclusive events whose union 1 2 n is the sample space S, i.e., one of the events must occur. Then if A is any event, we have the important theorem: Theorem 1-10 (Bayes’ Rule): P A( k )P A( | Ak ) P A( k | A) = (18) n ∑P A( j )P A( | Aj ) j=1 This enables us to find the probabilities of the various events A , 1 A , … , A that can occur. For this reason Bayes’ theorem is often 2 n referred to as a theorem on the probability of causes. Combinatorial Analysis In many cases the number of sample points in a sample space is not very large, and so direct enumeration or counting of sample points needed to obtain probabilities is not difficult. However, problems arise where direct counting becomes a practical impos- sibility. In such cases use is made of combinatorial analysis, which could also be called a sophisticated way of counting.
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10 PROBABILITY AND STATISTICS Fundamental Principle of Counting If one thing can be accomplished n different ways and after this a sec- 1 ond thing can be accomplished n different ways, … , and finally a kth 2 thing can be accomplished in n different ways, then all k things can be k accomplished in the specified order in n n …n different ways. 1 2 k Permutations Suppose that we are given n distinct objects and wish to arrange r of these objects in a line. Since there are n ways of choosing the first object, and after this is done, n – 1 ways of choosing the second object, … , and finally n – r + 1 ways of choosing the rth object, it follows by the fundamental principle of counting that the number of different arrangements, or permutations as they are often called, is given by n Pr = n n( −1)...(n − r +1) (19) where it is noted that the product has r factors. We call P the number n r of permutations of n objects taken r at a time. Example 1.4. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible? The men may be seated in P ways, and the women P ways. Each 5 5 4 4 arrangement of the men may be associated with each arrangement of the women. Hence, Number of arrangements = P , P = 5! 4! = (120)(24) = 2880 5 5 4 4 In the particular case when r = n, this becomes n Pn = n n( −1)(n − 2)...1= n! (20)
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CHAPTER 1: Basic Probability 11 which is called n factorial. We can write this formula in terms of facto- rials as n! n Pr = (21) (n − r)! If r = n, we see that the two previous equations agree only if we have 0! = 1, and we shall actually take this as the definition of 0!. Suppose that a set consists of n objects of which n are of one 1 type (i.e., indistinguishable from each other), n are of a second type, … , 2 n are of a kth type. Here, of course, n = n1 + n2 +...+ nk . Then the k number of different permutations of the objects is n! n P = (22) n1,n2 ,...,nk n !n !Ln ! 1 2 k Combinations In a permutation we are interested in the order of arrangements of the objects. For example, abc is a different permutation from bca. In many problems, however, we are only interested in selecting or choosing objects without regard to order. Such selections are called combina- tions. For example, abc and bca are the same combination. The total number of combinations of r objects selected from n (also called the combinations of n things taken r at a time) is denoted by C n r or n . We have r n n! =nCr = (23) r r!(n − r)! It can also be written n n n( −1)L(n − r +1) P = = n r (24) r r! r! It is easy to show that
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12 PROBABILITY AND STATISTICS n n r = n − r or C = C (25) n r n n−r Example 1.5. From 7 consonants and 5 vowels, how many words can be formed consisting of 4 different consonants and 3 different vow- els? The words need not have meaning. The four different consonants can be selected in C ways, the three dif- 7 4 ferent vowels can be selected in C ways, and the resulting 7 different 5 3 letters can then be arranged among themselves in P = 7! ways. Then 7 7 Number of words = C · C · 7! = 35·10·5040 = 1,764,000 7 4 5 3 Binomial Coefficients The numbers from the combinations formula are often called binomial coefficients because they arise in the binomial expansion n n n (x + y)n = xn + xn−1y + xn−2y2 +L+ yn (26) 1 2 n Stirling’s Approximation to n! When n is large, a direct evaluation of n! may be impractical. In such cases, use can be made of the approximate formula n ~ 2πn n en −n (27) where e = 2.71828 … , which is the base of natural logarithms. The symbol ~ means that the ratio of the left side to the right side approach- es 1 as n → ∞.
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