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AN INTRODUCTION TO PROBABILITY AND STATISTICS
by A.K. Md. Ehsanes Saleh, Vijay K. Rohatgi
An Introduction to Probability and Statistics, 3rd Edition
COVER
TITLE PAGE
PREFACE TO THE THIRD EDITION
PREFACE TO THE SECOND EDITION
PREFACE TO THE FIRST EDITION
ACKNOWLEDGMENTS
ENUMERATION OF THEOREMS AND REFERENCES
1 PROBABILITY
1.1 INTRODUCTION
1.2 SAMPLE SPACE
1.3 PROBABILITY AXIOMS
1.4 COMBINATORICS: PROBABILITY ON FINITE SAMPLE SPACES
1.5 CONDITIONAL PROBABILITY AND BAYES THEOREM
1.6 INDEPENDENCE OF EVENTS
2 RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS
2.1 INTRODUCTION
2.2 RANDOM VARIABLES
2.3 PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE
2.4 DISCRETE AND CONTINUOUS RANDOM VARIABLES
2.5 FUNCTIONS OF A RANDOM VARIABLE
3 MOMENTS AND GENERATING FUNCTIONS
3.1 INTRODUCTION
3.2 MOMENTS OF A DISTRIBUTION FUNCTION
3.3 GENERATING FUNCTIONS
3.4 SOME MOMENT INEQUALITIES
4 MULTIPLE RANDOM VARIABLES
4.1 INTRODUCTION
4.2 MULTIPLE RANDOM VARIABLES
4.3 INDEPENDENT RANDOM VARIABLES
4.4 FUNCTIONS OF SEVERAL RANDOM VARIABLES
4.5 COVARIANCE, CORRELATION AND MOMENTS
4.6 CONDITIONAL EXPECTATION
4.7 ORDER STATISTICS AND THEIR DISTRIBUTIONS
5 SOME SPECIAL DISTRIBUTIONS
5.1 INTRODUCTION
5.2 SOME DISCRETE DISTRIBUTIONS
5.3 SOME CONTINUOUS DISTRIBUTIONS
5.4 BIVARIATE AND MULTIVARIATE NORMAL DISTRIBUTIONS
5.5 EXPONENTIAL FAMILY OF DISTRIBUTIONS
6 SAMPLE STATISTICS AND THEIR DISTRIBUTIONS
6.1 INTRODUCTION
6.2 RANDOM SAMPLING
6.3 SAMPLE CHARACTERISTICS AND THEIR DISTRIBUTIONS
6.4 CHI-SQUARE, t-, AND F-DISTRIBUTIONS: EXACT SAMPLING DISTRIBUTIONS
6.5 DISTRIBUTION OF IN SAMPLING FROM A NORMAL POPULATION
6.6 SAMPLING FROM A BIVARIATE NORMAL DISTRIBUTION
7 BASIC ASYMPTOTICS: LARGE SAMPLE THEORY
7.1 INTRODUCTION
7.2 MODES OF CONVERGENCE
7.3 WEAK LAW OF LARGE NUMBERS
7.4 STRONG LAW OF LARGE NUMBERS†
7.5 LIMITING MOMENT GENERATING FUNCTIONS
7.6 CENTRAL LIMIT THEOREM
7.7 LARGE SAMPLE THEORY
8 PARAMETRIC POINT ESTIMATION
8.1 INTRODUCTION
8.2 PROBLEM OF POINT ESTIMATION
8.3 SUFFICIENCY, COMPLETENESS AND ANCILLARITY
8.4 UNBIASED ESTIMATION
8.5 UNBIASED ESTIMATION (CONTINUED): A LOWER BOUND FOR THE VARIANCE OF AN ESTIMATOR
8.6 SUBSTITUTION PRINCIPLE (METHOD OF MOMENTS)
8.7 MAXIMUM LIKELIHOOD ESTIMATORS
8.8 BAYES AND MINIMAX ESTIMATION
8.9 PRINCIPLE OF EQUIVARIANCE
9 NEYMAN–PEARSON THEORY OF TESTING OF HYPOTHESES
9.1 INTRODUCTION
9.2 SOME FUNDAMENTAL NOTIONS OF HYPOTHESES TESTING
9.3 NEYMAN–PEARSON LEMMA
9.4 FAMILIES WITH MONOTONE LIKELIHOOD RATIO
9.5 UNBIASED AND INVARIANT TESTS
9.6 LOCALLY MOST POWERFUL TESTS
10 SOME FURTHER RESULTS ON HYPOTHESES TESTING
10.1 INTRODUCTION
10.2 GENERALIZED LIKELIHOOD RATIO TESTS
10.3 CHI-SQUARE TESTS
10.4 t-TESTS
10.5 F-TESTS
10.6 BAYES AND MINIMAX PROCEDURES
11 CONFIDENCE ESTIMATION
11.1 INTRODUCTION
11.2 SOME FUNDAMENTAL NOTIONS OF CONFIDENCE ESTIMATION
11.3 METHODS OF FINDING CONFIDENCE INTERVALS
11.4 SHORTEST-LENGTH CONFIDENCE INTERVALS
11.5 UNBIASED AND EQUIVARIANT CONFIDENCE INTERVALS
11.6 RESAMPLING: BOOTSTRAP METHOD
12 GENERAL LINEAR HYPOTHESIS
12.1 INTRODUCTION
12.2 GENERAL LINEAR HYPOTHESIS
12.3 REGRESSION ANALYSIS
12.4 ONE-WAY ANALYSIS OF VARIANCE
12.5 TWO-WAY ANALYSIS OF VARIANCE WITH ONE OBSERVATION PER CELL
12.6 TWO-WAY ANALYSIS OF VARIANCE WITH INTERACTION
13 NONPARAMETRIC STATISTICAL INFERENCE
13.1 INTRODUCTION
13.2 U-STATISTICS
13.3 SOME SINGLE-SAMPLE PROBLEMS
13.4 SOME TWO-SAMPLE PROBLEMS
13.5 TESTS OF INDEPENDENCE
13.6 SOME APPLICATIONS OF ORDER STATISTICS
13.7 ROBUSTNESS
FREQUENTLY USED SYMBOLS AND ABBREVIATIONS
REFERENCES
STATISTICAL TABLES
ANSWERS TO SELECTED PROBLEMS
AUTHOR INDEX
SUBJECT INDEX
WILEY SERIES IN PROBABILITY AND STATISTICS
END USER LICENSE AGREEMENT
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COVER
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An Introduction to Probability and Statistics
Table of Contents
COVER
TITLE PAGE
PREFACE TO THE THIRD EDITION
PREFACE TO THE SECOND EDITION
PREFACE TO THE FIRST EDITION
ACKNOWLEDGMENTS
ENUMERATION OF THEOREMS AND REFERENCES
1 PROBABILITY
1.1 INTRODUCTION
1.2 SAMPLE SPACE
1.3 PROBABILITY AXIOMS
1.4 COMBINATORICS: PROBABILITY ON FINITE SAMPLE SPACES
1.5 CONDITIONAL PROBABILITY AND BAYES THEOREM
1.6 INDEPENDENCE OF EVENTS
2 RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS
2.1 INTRODUCTION
2.2 RANDOM VARIABLES
2.3 PROBABILITY DISTRIBUTION OF A RANDOM VARIABLE
2.4 DISCRETE AND CONTINUOUS RANDOM VARIABLES
2.5 FUNCTIONS OF A RANDOM VARIABLE
3 MOMENTS AND GENERATING FUNCTIONS
3.1 INTRODUCTION
3.2 MOMENTS OF A DISTRIBUTION FUNCTION
3.3 GENERATING FUNCTIONS
3.4 SOME MOMENT INEQUALITIES
4 MULTIPLE RANDOM VARIABLES
4.1 INTRODUCTION
4.2 MULTIPLE RANDOM VARIABLES
4.3 INDEPENDENT RANDOM VARIABLES
4.4 FUNCTIONS OF SEVERAL RANDOM VARIABLES
4.5 COVARIANCE, CORRELATION AND MOMENTS
4.6 CONDITIONAL EXPECTATION
4.7 ORDER STATISTICS AND THEIR DISTRIBUTIONS
5 SOME SPECIAL DISTRIBUTIONS
5.1 INTRODUCTION
5.2 SOME DISCRETE DISTRIBUTIONS
5.3 SOME CONTINUOUS DISTRIBUTIONS
5.4 BIVARIATE AND MULTIVARIATE NORMAL DISTRIBUTIONS
5.5 EXPONENTIAL FAMILY OF DISTRIBUTIONS
6 SAMPLE STATISTICS AND THEIR DISTRIBUTIONS
6.1 INTRODUCTION
6.2 RANDOM SAMPLING
6.3 SAMPLE CHARACTERISTICS AND THEIR DISTRIBUTIONS
6.4 CHI-SQUARE,
t
-, AND
F
-DISTRIBUTIONS: EXACT SAMPLING DISTRIBUTIONS
6.5 DISTRIBUTION OF
IN SAMPLING FROM A NORMAL POPULATION
6.6 SAMPLING FROM A BIVARIATE NORMAL DISTRIBUTION
7 BASIC ASYMPTOTICS: LARGE SAMPLE THEORY
7.1 INTRODUCTION
7.2 MODES OF CONVERGENCE
7.3 WEAK LAW OF LARGE NUMBERS
7.4 STRONG LAW OF LARGE NUMBERS
†
7.5 LIMITING MOMENT GENERATING FUNCTIONS
7.6 CENTRAL LIMIT THEOREM
7.7 LARGE SAMPLE THEORY
8 PARAMETRIC POINT ESTIMATION
8.1 INTRODUCTION
8.2 PROBLEM OF POINT ESTIMATION
8.3 SUFFICIENCY, COMPLETENESS AND ANCILLARITY
8.4 UNBIASED ESTIMATION
8.5 UNBIASED ESTIMATION (
CONTINUED
): A LOWER BOUND FOR THE VARIANCE OF AN ESTIMATOR
8.6 SUBSTITUTION PRINCIPLE (METHOD OF MOMENTS)
8.7 MAXIMUM LIKELIHOOD ESTIMATORS
8.8 BAYES AND MINIMAX ESTIMATION
8.9 PRINCIPLE OF EQUIVARIANCE
9 NEYMAN–PEARSON THEORY OF TESTING OF HYPOTHESES
9.1 INTRODUCTION
9.2 SOME FUNDAMENTAL NOTIONS OF HYPOTHESES TESTING
9.3 NEYMAN–PEARSON LEMMA
9.4 FAMILIES WITH MONOTONE LIKELIHOOD RATIO
9.5 UNBIASED AND INVARIANT TESTS
9.6 LOCALLY MOST POWERFUL TESTS
10 SOME FURTHER RESULTS ON HYPOTHESES TESTING
10.1 INTRODUCTION
10.2 GENERALIZED LIKELIHOOD RATIO TESTS
10.3 CHI-SQUARE TESTS
10.4
t
-TESTS
10.5
F
-TESTS
10.6 BAYES AND MINIMAX PROCEDURES
11 CONFIDENCE ESTIMATION
11.1 INTRODUCTION
11.2 SOME FUNDAMENTAL NOTIONS OF CONFIDENCE ESTIMATION
11.3 METHODS OF FINDING CONFIDENCE INTERVALS
11.4 SHORTEST-LENGTH CONFIDENCE INTERVALS
11.5 UNBIASED AND EQUIVARIANT CONFIDENCE INTERVALS
11.6 RESAMPLING: BOOTSTRAP METHOD
12 GENERAL LINEAR HYPOTHESIS
12.1 INTRODUCTION
12.2 GENERAL LINEAR HYPOTHESIS
12.3 REGRESSION ANALYSIS
12.4 ONE-WAY ANALYSIS OF VARIANCE
12.5 TWO-WAY ANALYSIS OF VARIANCE WITH ONE OBSERVATION PER CELL
12.6 TWO-WAY ANALYSIS OF VARIANCE WITH INTERACTION
13 NONPARAMETRIC STATISTICAL INFERENCE
13.1 INTRODUCTION
13.2
U
-STATISTICS
13.3 SOME SINGLE-SAMPLE PROBLEMS
13.4 SOME TWO-SAMPLE PROBLEMS
13.5 TESTS OF INDEPENDENCE
13.6 SOME APPLICATIONS OF ORDER STATISTICS
13.7 ROBUSTNESS
FREQUENTLY USED SYMBOLS AND ABBREVIATIONS
REFERENCES
STATISTICAL TABLES
ANSWERS TO SELECTED PROBLEMS
AUTHOR INDEX
SUBJECT INDEX
WILEY SERIES IN PROBABILITY AND STATISTICS
END USER LICENSE AGREEMENT
List of Tables
Statistical Tables
Table ST1. Cumulative Binomial Probabilities,
,
r
= 0,1,2,…,
n
− 1
Table ST2. Tail Probability Under Standard Normal Distribution
Table ST3. Critical Values Under Chi-Square Distribution
Table ST4. Student's
t
-Distribution
Table ST5.
F
-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the Distribution of
F
Table ST6. Random Normal Numbers,
μ
= 0 and
σ
= 1
Table ST7. Critical Values of the Kolmogorov-Smirnov One-Sample Test Statistic
Table ST8. Critical Values of the Kolmogorov-Smirnov Test Statistic for Two Samples of Equal Size
Table ST9. Critical Values of the Kolmogorov-Smirnov Test Statistic for Two Samples of Unequal Size
Table ST10. Critical Values of the Wilcoxon Signed-Ranks Test Statistic
Table ST11. Critical Values of the Mann-Whitney-Wilcoxon Test Statistic
Table ST12. Critical Points of Kendall's Tau Test Statistic
Table ST13. Critical Values of Spearman's Rank Correlation Statistic
List of Illustrations
Chapter 01
Fig. 1
Fig. 2
Fig. 1
A
= {(
x
,
y
): 0 ≤
x
≤ 1/2, 1/2 ≤
y
≤ 1}.
Fig. 2
B
=
{
(
x
,
y
):(
x
- 1 /2)
2
+ (
y
- 1 /2)
2
= 1
}
.
Fig. 3
C
= {(
x
,
y
) : (
x
2
+
y
2
≤ 1}
Fig. 4 {(
x
,
y
) : 0 <
x
< 1/2 <
y
< 1, and (
y
−
x
) < 1/2 or 0 <
y
< 1/2 <
x
< 1, and (
x
−
y
) < 1/2}.
Fig. 5 {(
x,y
): 0 <
x
<1/2, 1/2 <
y
<1and 2 (
y
-x
)
<1}.
Fig. 6
Fig. 7
Fig. 8
Fig. 1 Map for Problem 11.
Chapter 02
Fig. 1
Fig. 2
Fig. 3 Graph of
f
.
Fig. 4 Graph of
F
.
Fig. 1
.
Fig. 2
,
.
Chapter 03
Fig. 1 Quantile of order
p
.
Fig. 2 (a) Unique quantile and (b) infinitely many solutions of
F
(
x
) =
p
.
Fig. 1 Chebychev upper bound versus exact probability
Chapter 04
Fig. 1
Fig. 2
Fig. 3
.
Fig. 4
.
Fig. 1 (a)
and (b)
.
Fig. 2
.
Fig. 3
.
Chapter 05
Fig. 1 (
a
) Exponential location family; (
b
) exponential scale family; (
c
) normal location-scale family; and (
d
) shaped parameter family
.
Fig. 2 Gamma density functions.
Fig. 3 Beta density functions.
Fig. 4 Cauchy density function.
Fig. 1 Bivariate normal with
,
and
, −0.5,0.5,0.9.
Chapter 06
Fig. 1 Empirical DF for data of Example 1.
Fig. 2
.
Fig. 1 Chi-square densities.
Fig. 2 Student’s
t
-densities
Fig. 3
Fig. 4
F
densities.
Chapter 07
Fig. 1 (
a
) Distribution of
for Poisson RV with mean 3 and normal approximation and (
b
) distribution of
for exponential RV with mean 1 and normal approximation.
Chapter 08
Fig. 1 Comparison of
R
(
p
,
δ
) and
.
Fig. 2 Comparison of
R
(
p
,
δ
) and
.
Chapter 09
Fig. 1 Rejection region of
H
0
in Example 5.
Fig. 2 Power function of
φ
in Example 5.
Fig. 3 Power function of
φ
in Example 6.
Fig. 1
Fig. 2 Graph of
Fig. 1 Power functions of Chi-square tests of
against
H
1
.
Chapter 11
Fig. 1
Guide
Cover
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