Schaum's Outline of Probability and Statistics, 4th Edition: 897 Solved Problems + 20 Videos - Softcover

SCHILLER Jr., John J.

Softcover

ISBN 10: 007179557X ISBN 13: 9780071795579

Publisher: McGraw Hill, 2012

Synopsis

Tough Test Questions? Missed Lectures? Not Enough Time?

Fortunately, there's Schaum's. This all-in-one-package includes more than 750 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.

This Schaum's Outline gives you

• 897 fully solved problems
• Concise explanations of all course fundamentals
• Information on conditional probability and independence, random variables, binominal and normal distributions, sampling distributions, and analysis of variance

Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum’s to shorten your study time--and get your best test scores!

Schaum's Outlines--Problem Solved.

About this title

About the Authors

John J. Schiller, is an Associate Professor of Mathematics at Temple University. He received his Ph.D. at the University of Pennsylvania and has published research papers in the areas of Riemann surfaces, discrete mathematics biology. He has also coauthored texts in finite mathematics, precalculus, and calculus.

Alu Srinivasan is Professor of Mathematics at Temple University. He received an M.S. in statistics and a Ph.D. in Mathematics from Wayne State University, and was Chair of Temple's Mathematics Department from 1990 to 1998. His primary research interests are in applied and mathematical statistics, combinatorics and probability. He has published some seventy papers in these areas and has supervised a dozen Ph.D. dissertations in statistical inference and biostatistics.

The Late MURRAY R. SPIEGEl received the M.S degree in Physics and the Ph.D. in Mathematics from Cornell University. He had positions at Harvard University, Columbia University, Oak Ridge and Rensselaer Polytechnic Insitute, and served as a mathematical consultant at several large Companies. His last Position was professor and Chairman of mathematics at the Rensselaer Polytechnic Institute Hartford Graduate Center. He was interested in most branches of mathematics at the Rensselaer polytechnic Institute, Hartford Graduate Center. He was interested in most branches of mathematics, especially those which involve applications to physics and engineering problems. He was the author of numerous journal articles and 14 books on various topics in mathematics.

Excerpt. © Reprinted by permission. All rights reserved.

Probability and Statistics

By Murray R. Spiegel John J. Schiller R. Alu Srinivasan

McGraw-Hill Companies, Inc.

Copyright © 2013 The McGraw-Hill Companies, Inc.
All right reserved.
ISBN: 978-0-07-179557-9

Contents

Chapter One

Basic Probability

Random Experiments

We are all familiar with the importance of experiments 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 control 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 conditions are the same. These experiments are described as random. The following are some examples.

EXAMPLE 1.1 If we toss a coin, the result of the experiment is that it will either come up "tails," symbolized by T (or 0), or "heads," symbolized by H (or 1), i.e., one of the elements of the set {H, T} (or {0, 1}).

EXAMPLE 1.2 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}.

EXAMPLE 1.3 If we toss a coin twice, there are four results possible, as indicated by {HH, HT, TH, TT}, i.e., both heads, heads on first and tails on second, etc.

EXAMPLE 1.4 If we are making bolts with a machine, the result of the experiment is that some may be defective. Thus when a bolt is made, it will be a member of the set {defective, nondefective}.

EXAMPLE 1.5 If an experiment consists of measuring "lifetimes" of electric light bulbs produced by a company, then the result of the experiment is a time t in hours that lies in some interval—say, 0 [less than or equal to] t [less than or equal to] 4000—where we assume that no bulb lasts more than 4000 hours.

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 1.6 If we toss a die, one sample space, or set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6} while another is {odd, even}. It is clear, however, that the latter would not be adequate to determine, for example, whether an outcome is divisible by 3.

It is often useful to portray a sample space graphically. In such cases it is desirable to use numbers in place of letters whenever possible.

EXAMPLE 1.7 If we toss a coin twice and use 0 to represent tails and 1 to represent heads, the sample space (see Example 1.3) can be portrayed by points as in Fig. 1-1 where, for example, (0, 1) represents tails on first toss and heads on second toss, i.e., TH.

If a sample space has a finite number of points, as in Example 1.7, 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 [less than or equal to] x [less than or equal to] 1, it is called a noncountably infinite sample space. A sample space that is finite or countably infinite is often called a discrete sample space, while one that is noncountably infinite is called 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 often called a simple or elementary event.

EXAMPLE 1.8 If we toss a coin twice, the event that only one head comes up is the subset of the sample space that consists of points (0, 1) and (1, 0), as indicated in Fig. 1- 2.

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 [empty set], which is called the impossible event because an element of [empty set] 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 [union] B is the event "either A or B or both." A [union] B is called the union of A and B.

2. A [intersection] B is the event "both A and B." A [intersection] B is called the intersection of A and B.

3. A' is the event "not A." A' is called the complement of A.

4. A - B = A [intersection] 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 [intersection] B = [empty set], we often say that the events are mutually exclusive. This means that they cannot both occur. We say that a collection of events A₁, A₂, ..., A_n is mutually exclusive if every pair in the collection is mutually exclusive.

EXAMPLE 1.9 Referring to the experiment of tossing a coin twice, let A be the event "at least one head occurs" and B the event "the second toss results in a tail." Then A = {HT, TH, HH}, B = {HT, TT}, and so we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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 convenient to assign a number between 0 and 1. If we are sure or certain that the event will occur, we say that its probability is 100% or 1, but if we are sure that the event will not occur, we say that its probability is zero. If, for example, the probability is ¼ 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 its occurrence are 75% to 25%, or 3 to 1.

There are two important procedures by means of which we can estimate the probability of an event.

1. CLASSICAL APPROACH. If an event can occur in h different ways out of a total number of n possible ways, all of which are equally likely, then the probability of the event is h/n.

EXAMPLE 1.10 Suppose we want to know the probability that a head will turn up in a single toss of a coin. Since there are two equally likely ways in which the coin can come up—namely, heads and tails (assuming it does not roll away or stand on its edge)—and of these two ways a head can arise in only one way, we reason that the required probability is 1/2. In arriving at this, we assume that the coin is fair, i.e., not loaded in any way.

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.

EXAMPLE 1.11 If we toss a coin 1000 times and find that it comes up heads 532 times, we estimate the probability of a head coming up to be 532/1000 = 0.532.

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 correspond to events and conversely, but if S is nondiscrete, only special subsets (called measurable) correspond to events. To each event A in the class ITLITL of events, we associate a real number P(A). Then P is called a probability function, and P(A) the probability of the event A, if the following axioms are satisfied.

Axiom 1 For every event A in the class ITLITL,

P(A) [greater than or equal to] 0 (1)

Axiom 2 For the sure or certain event S in the class ITLITL,

P(S) = 1 (2)

Axiom 3 For any number of mutually exclusive events A₁, A₂, ..., in the class ITLITL,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

In particular, for two mutually exclusive events A₁, A₂,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Some Important Theorems on Probability

From the above axioms we can now prove various theorems on probability that are important in further work.

Theorem 1-1 If A₁ (A₂, then P(A₁) [less than or equal to] P(A₂) and P(A₂ - A₁) = P(A₂) - P(A₁)

Theorem 1-2 For every event A,

0 [less than or equal to] P(A) [less than or equal to] 1, (5)

i.e., a probability is between 0 and 1.

Theorem 1-3

P([empty set]) = 0 (6)

i.e., the impossible event has probability zero.

Theorem 1-4 If A' is the complement of A, then

P(A') = 1 - P(A) (7)

Theorem 1-5 If A = A₁ [union] A₂ [union] ··· [union] A_n, where A₁, A₂, ..., A_n are mutually exclusive events, then

P(A) = P(A₁) + P(A₂) + ··· + P(A_N) (8)

In particular, if A = S, the sample space, then

P(A₁) + P(A₂) + ··· + P(A_N) = 1 (9)

Theorem 1-6 If A and B are any two events, then

P(A [union] B) = P(A) + P(B) - P(A [intersection] B) (10)

More generally, if A₁, A₂, A₃ are any three events, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Generalizations to n events can also be made.

Theorem 1-7 For any events A and B,

P(A) = P(A [intersection] B) + P(A [intersection] B') (12)

Theorem 1-8 If an event A must result in the occurrence of one of the mutually exclusive events A₁, A₂, ..., A_n, then

P(A) = P(A [intersection] A₁) + P(A [intersection] A₂) + ··· + P(A [intersection] A_n) (13)

Assignment of Probabilities

If a sample space S consists of a finite number of outcomes a₁, a₂, ..., a_n, then by Theorem 1-5,

P(A₁) + P(A₂) + ··· + P(A_n) = 1 (14)

where A₁, A₂, ..., A_n are elementary events given by A_i = {a_i}.

It follows that we can arbitrarily choose any nonnegative numbers for the probabilities of these simple events as long as (14) is satisfied. In particular, if we assume equal probabilities for all simple events, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and if A is any event made up of h such simple events, we have

P(A) = h/n (16)

This is equivalent to the classical approach to probability given on page 5. We could of course use other procedures for assigning probabilities, such as the frequency approach of page 5.

Assigning probabilities provides a mathematical model, the success of which must be tested by experiment in much the same manner that theories in physics or other sciences must be tested by experiment.

EXAMPLE 1.12 A single die is tossed once. Find the probability of a 2 or 5 turning up.

The sample space is S = (1, 2, 3, 4, 5, 6). If we assign equal probabilities to the sample points, i.e., if we assume that the die is fair, then

P(1) = P(2) = ··· = P(6) = 1/6

The event that either 2 or 5 turns up is indicated by 2 [union] 5. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Conditional Probability

Let A and B be two events (Fig. 1-3) such that P(A) > 0. Denote by P(B | A) the probability 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

In words, (18) says 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 probability 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 on page 5.

EXAMPLE 1.13 Find the probability that a single toss of a die will result in a number less than 4 if (a) no other information is given and (b) it is given that the toss resulted in an odd number.

(a) Let B denote the event {less than 4}. Since B is the union of the events 1, 2, or 3 turning up, we see by Theorem 1-5 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

assuming equal probabilities for the sample points.

(b) Letting A be the event {odd number}, we see that P(A) = 3/6 = ½. Also P(A [intersection] B) = 2/6 = 1/3. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the added knowledge that the toss results in an odd number raises the probability from 1/2 to 2/3.

Theorems on Conditional Probability

Theorem 1-9 For any three events A₁, A₂, A₃, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

(Continues...)

Excerpted from Probability and Statisticsby Murray R. Spiegel John J. Schiller R. Alu Srinivasan Copyright © 2013 by The McGraw-Hill Companies, Inc.. Excerpted by permission of McGraw-Hill Companies, Inc.. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Edition details

• Publisher: McGraw Hill
• Publication date: 2012
• ISBN 10: 007179557X
• ISBN 13: 9780071795579
• Binding: Paperback
• Language: English
• Edition number: 4
• Number of pages: 434