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, ..., An 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
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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...
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