26
Chapter 2
Probability
2.4
If A and B are two sets, draw Venn diagrams to verify the following:
a
b
2.5
2.6
A = (A B) (A B).
If B A then A = B (A B).
Refer to Exercise 2.4. Use the identities A = A S and S = B B and a distributive law to
prove that
a A
CHAPTER
1
What Is Statistics?
1.1 Introduction
1.2 Characterizing a Set of Measurements: Graphical Methods
1.3 Characterizing a Set of Measurements: Numerical Methods
1.4 How Inferences Are Made
1.5 Theory and Reality
1.6 Summary
References and Further Re
Contents vii
4.8
Some General Comments
4.9
Other Expected Values
4.10
Tchebysheffs Theorem
4.11
Expectations of Discontinuous Functions and Mixed Probability
Distributions (Optional) 210
4.12
Summary
201
202
207
214
5 Multivariate Probability Distribution
1.3
Characterizing a Set of Measurements: Numerical Methods
9
could be formed from the same set of measurements. To make inferences about a
population based on information contained in a sample and to measure the goodness
of the inferences, we need rigoro
32
Chapter 2
Probability
e Because A = E 8 E 9 E 10 , Axiom 3 implies that
P(A) = P(E 8 ) + P(E 9 ) + P(E 10 ) = 3/10.
The next section contains an axiomatic description of the method for calculating
P(A) that we just used.
Before we proceed, let us note
2
Chapter 1
What Is Statistics?
problem of making decisions in the face of uncertainty. And Mood, Graybill, and
Boes (1974) define statistics as the technology of the scientific method and add
that statistics is concerned with (1) the design of experiment
NOTE TO THE STUDENT
As the title Mathematical Statistics with Applications implies, this text is concerned
with statistics, in both theory and application, and only deals with mathematics as a
necessary tool to give you a firm understanding of statistical
Preface xv
FIGURE 1
Applet illustration of
Bayes rule
random variable are equivalent to probabilities associated with the standard normal
distribution. The applet Normal Probabilities (One Tail) provides upper-tail areas associated with any user-specified
xii
Contents
A1.5
Identity Elements
A1.6
The Inverse of a Matrix
A1.7
The Transpose of a Matrix
A1.8
A Matrix Expression for a System of Simultaneous
Linear Equations 828
A1.9
Inverting a Matrix
A1.10
Solving a System of Simultaneous Linear Equations
A1.1
CONTENTS
Preface
xiii
Note to the Student
xxi
1 What Is Statistics? 1
1.1
Introduction
1.2
Characterizing a Set of Measurements: Graphical Methods
3
1.3
Characterizing a Set of Measurements: Numerical Methods
8
1.4
How Inferences Are Made
1.5
Theory and R
Contents xi
14.4
Contingency Tables
14.5
r c Tables with Fixed Row or Column Totals
14.6
14.7
Other Applications
721
729
734
Summary and Concluding Remarks
736
15 Nonparametric Statistics 741
15.1
Introduction
15.2
A General Two-Sample Shift Model
15.3
Th
xviii
Preface
illustration of convergence in probability. In Chapter 10, the applet Hypothesis Testing
(for Proportions) illustrates important concepts associated with test of hypotheses
including the following:
What does really mean?
Tests based on lar
10
Chapter 1
What Is Statistics?
DEFINITION 1.2
The variance of a sample of measurements y1 , y2 , . . . , yn is the sum of the
square of the differences between the measurements and their mean, divided
by n 1. Symbolically, the sample variance is
s2 =
n
18
Chapter 1
What Is Statistics?
Munich. Of these 2000 women, the 48% who worked outside the home had HDL levels that were
between 2.5 and 3.6 milligrams per deciliter (mg/dL) higher than the HDL levels of their stayat-home counterparts. Suppose that the
Exercises
7
c If one of the stocks is selected at random from the 40 for which the preceding data were
taken, what is the probability that it will have traded fewer than 5% of its outstanding shares?
1.5
Given here is the relative frequency histogram asso
10
Integers and Equivalence Relations
An item with the UPC identification number a1a2 ? a12 satisfies the
condition
(a1, a2, . . . , a12) ? (3, 1, 3, 1, . . . , 3, 1) mod 10 5 0.
To verify that the number in Figure 0.4 satisfies the condition above, we
ca
vi
Contents
Computer Exercises 273
Biography of Richard Dedekind 274
Biography of Emmy Noether 275
Supplementary Exercises for Chapters 1214 276
15 Ring Homomorphisms 280
Definition and Examples 280 | Properties of Ring Homomorphisms
283 | The Field of Qu
18
Integers and Equivalence Relations
Theorem 0.6 Equivalence Classes Partition
The equivalence classes of an equivalence relation on a set S
constitute a partition of S. Conversely, for any partition P of S, there
is an equivalence relation on S whose eq
Exercises
Opinion
33
Proportion
Very likely
Somewhat likely
Unlikely
Other
.24
.24
.40
.12
Suppose that one American is selected and his or her opinion is recorded.
a What are the simple events for this experiment?
b Are the simple events that you gave in
30
Chapter 2
Probability
2. The relative frequency of the whole sample space S must be unity. Because
every possible outcome of the experiment is a point in S, it follows that S must
occur every time the experiment is performed.
3. If two events are mutua
1.4
How Inferences Are Made
13
1.20
Weekly maintenance costs for a factory, recorded over a long period of time and adjusted
for inflation, tend to have an approximately normal distribution with an average of $420 and a
standard deviation of $30. If $450
28
Chapter 2
Probability
F I G U R E 2.7
Venn diagram for the
sample space
associated with
the die-tossing
experiment
DEFINITION 2.3
S
E6
E1
E5
E3
E2
E4
The sample space associated with an experiment is the set consisting of all
possible sample points. A
1.2
Characterizing a Set of Measurements: Graphical Methods
3
an inference about a population based on information contained in a sample from
that population and to provide an associated measure of goodness for the inference.
Exercises
1.1
For each of the
34
Chapter 2
Probability
selected from the records of the center. What is the probability that the person selected has
a type O+ blood?
b type O blood?
c type A blood?
d neither type A nor type O blood?
2.17
Hydraulic landing assemblies coming from an air
4
Chapter 1
What Is Statistics?
contract, fixed dollar costs, and the supervisor who is assigned the task of organizing
and conducting the manufacturing operation. The statistician will wish to measure the
response or dependent variable, profit per contra
Exercises
F I G U R E 2.6
Venn diagram for
mutually exclusive
sets A and B
25
S
A
B
We will not attempt a thorough review of set algebra, but we mention four equalities
of considerable importance. These are the distributive laws, given by
A (B C) = (A B)
iv
Contents
4 Cyclic Groups 72
Properties of Cyclic Groups 72 | Classification of Subgroups
of Cyclic Groups 77
Exercises 81
Computer Exercises 86
Biography of J. J. Sylvester 89
Supplementary Exercises for Chapters 14 91
5 Permutation Groups 95
Definitio
vi
Contents
2.9
Calculating the Probability of an Event: The Event-Composition
Method 62
2.10
The Law of Total Probability and Bayes Rule
2.11
Numerical Events and Random Variables
2.12
Random Sampling
2.13
Summary
70
75
77
79
3 Discrete Random Variables
Discrete Distributions
Distribution
Binomial
Probability Function
p(y) =
! "
n
y
p y (1 p)ny ;
Mean
Variance
MomentGenerating
Function
np
np(1 p)
[ pet + (1 p)]n
1
p
1 p
pet
1 (1 p)et
y = 0, 1, . . . , n
Geometric
p(y) = p(1 p) y1 ;
p2
y = 1, 2, . . .
Hyp
Preface xix
appendix. As in previous editions, some of the new exercises are theoretical whereas
others contain data from documented sources that deal with research in a variety of
fields. We continue to believe that exercises based on real data or actual