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# Fundamentals of Mathematical Statistics

575

Author: Veerarajan T

ISBN: 9789380381640

Copy Right Year: 2017

Pages:  842

Binding: Soft Cover

Publisher:  Yes Dee Publishing

SKU: 9789380381640 Category:

## Description

This book contains a balanced coverage of both the theory and applications that helps the beginners acquire a thorough knowledge of the concepts of mathematical statistics offered to the Arts, Science, Commerce and Engineering students in Indian Universities.

Weight 1.02 kg 23 × 18 × 3 cm

## Table of Content

Chapter 1 INTRODUCTION OF STATISTICS – PRESENTATION OF
STATISTICAL DATA
1.1 Scope of Statistics
1.2 Limitations of Statistics
1.3 Organization of Chapters in the Book
1.4 Classification of Raw Data and Frequency Distribution
1.5 Frequency Graphs
1.5.1 Line Diagram
1.5.2 Rectangular Histogram
1.5.3 Frequency Polygon
1.5.4 Frequency Curve
1.5.5 Cumulative Frequency Distributions and Ogives
Worked Examples 1
Exercise1
Chapter 2 MEASURES OF CENTRAL TENDENCY – AVERAGES
2.1 Averages
2.1.1 Arithmetic Mean (A.M.)
2.1.2 Properties of the Arithmetic Mean
2.2 Median
2.2.1 Formula for the Median of a Frequency Distribution
2.3 Mode
2.3.1 Empirical Relation between Mean, Median and Mode
2.4 Geometric Mean
2.5 Harmonic Mean
2.6 Requisites of a Satisfactory Average
2.7 Relative Merits and Demerits of Different Averages
Worked Examples 2
Exercise2
Chapter 3 MOMENTS, MEASURES OF DISPERSION, SKEWNESS
AND KURTOSIS
3.1 Moments – Definitions
3.2 Relations between Central and Raw Moments and Vice Versa
3.3 Computation of Higher Order Moments
3.4 Sheppard’s Correction for Moments
3.5 Dispersion
3.5.1 Range
3.5.2 Quartile Deviation
3.5.3 Mean Deviation
3.5.4 Minimal Property of Mean Deviation
3.6 Standard Deviation (S.D.)
3.6.1 Minimum Value of RMSD sA
3.6.2 Requisites of aSatisfactory Measure of Dispersion
3.6.3 Formula for the Computation of σ for a Frequency Distribution with Classes of Equal Class interval
3.6.4 Empirical Relation between Q.D., M.D. and S.D
3.6.5 S.D. of the Combination of Two Groups of Observations
3.6.6 An Alternative Formula for σ2
3.6.7 Relative Measures of Dispersion
3.7 Skewness
3.8 Kurtosis
3.9 Pearson’s β− and γ− Coefficients
Worked Examples 3
Exercise3
Chapter 4 THEORY OF PROBABILITY
4.1 Deterministic and Random Experiments
4.2 Different Ways of Defining Probability
4.2.1 Mathematical or Apriori Definition of Probability
4.2.2 Statistical or Aposteriori Definition of Probability
4.2.3 Axiomatic Definition of Probability
4.3 Some Theorems on Probability
4.4 Related and Independent Events
4.4.1 Conditional Probability and Product Theorem of Probability
4.4.2 Independent Events
4.4.3 Property
4.5 Theorem of Total Probability
4.6 Baye’s Theorem on Inverse Probability or the Theorem of Probability of Causes
4.7 Bernouilli’s Trials
4.8 Bernouilli’s Theorem
4.9 De Moivre-Laplace Approximation
4.10 Multinomial Probability – Extension of Bernouilli’s Theorem
Worked Examples 4(a)
Worked Examples 4(b)
Exercise 4(a)
Exercise 4(b)
Chapter 5 RANDOM VARIABLES
5.1 Discrete Random Variable
5.1.1 Probability Mass Function
5.1.2 Special Discrete Distributions
5.2 Continuous Random Variable
5.2.1 Probability Density Function
5.2.2 Special Continuous Distributions
5.3 Cumulative Distribution Function (C.D.F.)
5.3.1 Properties of the c.d.f. F(x)
5.4 Two-Dimensional Random Variables
5.4.1 Definitions
5.4.2 Joint Probability Mass Function of (X, Y )
5.4.3 Joint Probability Density Function of ( X, Y )
5.4.4 Cumulative Distribution Function of (X, Y )
5.4.5 Properties of F(x, y)
5.4.6 Marginal Probability Distribution/Density
5.4.7 Conditional Probability Distribution/Density
5.4.8 Independent Random Variables
5.4.9 Random Vectors or n-Dimensional R.V.’s
Worked Examples 5(a)
Worked Examples 5(b)
Exercise5(a)
Exercise5(b)
Chapter 6 TRANSFORMATION OF RANDOM VARIABLE
6.1 Transformation of one R.V. into another R.V
6.1.1 Probability Density Function of the R.V. Y , where Y = g (X), in terms of
the P.D.F. of X
6.2 Transformation from Two R.V.’s into a Single R.V
6.2.1 P.D.F. of Z = X + Y , where X and Y are Independent R.V.’s
6.2.2 P.D.F. of Z = XY , where X and Y are Independent R.V.’s
6.2.3 P.D.F. of Z = XY, where X and Y are Independent R.V.’s
6.3 Transformation of Two R.V.’s into Two Other R.V.’s
6.3.1 An Alternative Method to Find the p.d.f. of Z = g(X, Y )
Worked Examples 6
Exercise 6
Chapter 7 MATHEMATICAL EXPECTATION
7.1 Definitions of Mean and Variance
7.2 Elementary Properties of Mean and Variance
7.3 Expected value of a Function of a R.V
7.3.1 Other Statistical Measures for Continuous Probability Distributions
7.4 Expected Values of a Two Dimensional R.V
7.4.1 Property 1
7.4.2 Property 2
7.4.3 Property 3
7.4.4 Property 4
7.5 Covariance of X, Y
7.5.1 Properties of Covariance
7.6 Conditional Expected Values – Definitions
7.6.1 Property 1
7.6.2 Property 2
7.6.3 Property 3
7.6.4 Property 4
Worked Examples 7
Exercise7
Chapter 8 MOMENT GENERATING FUNCTION, INEQUALITIES AND
CONVERGENCE
8.1 Definitions
8.2 Properties of Characteristic Function
8.2.1 Property 1
8.2.2 Property 2
8.2.3 Property 3
8.2.4 Property 4
8.2.5 Property 5
8.2.6 Property 6
8.2.7 Property 7
8.3 Cumulant Generating Function (C.G.F.)
8.4 Joint Characteristic Function of a 2-Dimensional R.V
8.5 Probabilistic Inequalities
8.5.1 Tchebycheff Inequality
8.5.2 Markov’s Inequality
8.5.3 Bienayme’s Inequality
8.5.4 Cauchy-Schwartz Inequality
8.6 Convergence Concepts in Probability Theory
8.7 Central Limit Theorem(Lindeberg-Levy’s Form)
8.7.1 Central Limit Theorem (Liapounoff’s form)
Worked Examples 8(a)
Worked Examples 8(b)
Exercise8(a)
Exercise8(b)
Chapter 9 SPECIAL DISCRETE PROBABILITY DISTRIBUTIONS
9.1 Binomial Distribution
9.1.1 Characteristic Function of Binomial Distribution, Mean and Variance
9.1.2 Recurrence Formula for the Central Moments of the Binomial Distribution
and Values of μ2, μ3 and μ4
9.1.3 Mode of the Binomial Distribution
9.1.4 Cumulants of the Binomial Distribution
9.2 Poisson Distribution
9.2.1 Poisson Distribution as the Limiting Form of Binomial Distribution
9.2.2 Occurrence of Poisson Distribution
9.2.3 Moment Generating Function and Values of the First Four Central Moments of the Poisson Distribution
9.2.4 Recurrence Formula for the Central Moments of the Poisson
Distribution and Values of μ2, μ3 and μ4
9.2.5 Mode of the Poisson Distribution
9.2.6 Cumulants of the Poisson Distribution
9.2.7 Additive or Reproductive Property of Independent Poisson R.V.’s
9.3 Geometric Distribution
9.3.1 Characteristic Function, Mean and Variance of G*(p) Distribution
9.3.2 The First Four Cumulants and Central Moments of G*(p) Distribution . . . 357
9.3.3 Memoryless Property of G*(p) Distribution and its Converse
9.3.4 Median and Mode of the G*(p) Distribution
9.4 Negative Binomial Distribution
9.4.1 Moment Generating Function, Mean and Variance
9.4.2 Alternative Form of the P.M.F. of the Negative Binomial Distribution
9.4.3 The First Four Cumulants and Central Moments of the Negative Binomial
Distribution
9.4.4 Deduction of Central Moments of the Negative Binomial Distribution from Those of Binomial Distribution
9.4.5 Poisson Distribution as a Limiting Form of the Negative Binomial
Distribution
9.4.6 Recurrence Formula for the Central Moments of the Negative Binomial
Distribution
9.4.7 Reproductive Property of the Negative Binomial Distribution
9.5 Hypergeometric Distribution
9.5.1 Mean and Variance of the Hypergeometric Distribution
9.5.2 Binomial Distribution as Limiting Form of Hyper Geometric
Distribution
Worked Examples 9
Exercise 9
Chapter 10 SPECIAL CONTINUOUS PROBABILITY DISTRIBUTIONS
10.1 Uniform or Rectangular Distribution
10.1.1 Moment Generating Function and Mean, Variance, β1 and β2
10.1.2 The Absolute Moments and Mean Deviation of U(a, b)
10.2 Exponential Distribution
10.2.1 Moment Generating Function and Cumulant Generating Function
10.2.2 Memory less Property of the Exponential Distribution and its Converse
10.3 Erlang Distribution/General Gamma Distribution
10.3.1 Mean and Variance of General Gamma Distribution
10.3.2 M.G.F.,C.G.F. and Central Moments
10.3.3 Reproductive Property
10.3.4 Relation between the Distribution Functions of the Simple Gamma
Distribution with Parameter (k + 1) and Poisson Distribution with
Parameter λ
10.4 Weibull Distribution
10.4.1 Probability Density Function of the Weibull’s Distribution
10.4.2 Mean and Variance of Weibull Distribution
10.5 Double Exponential or Laplace Distribution
10.5.1 M.G.F. Mean and Variance of Two Parameter Laplace Distribution
10.6 Beta Distribution of the First Kind
10.6.1 Simple Moments and Harmonic Mean of the Beta I Distribution
10.7 Beta Distribution of the Second Kind
10.7.1 Simple Moments and Harmonic Mean of the Beta II Distribution
10.8 Cauchy Distribution
10.8.1 Characteristic Function of Cauchy’s Distribution
10.8.2 Mean, Median and Mode of the Two Parameter Cauchy Distribution
10.8.3 Reproductive Property of the General Cauchy Distribution
10.9 Normal (or Gaussian) Distribution
10.9.1 Normal Probability Curve and its Characteristics
10.9.2 Mean and Variance of the Normal Distribution N(μ, σ)
10.9.3 Median and Mode of the Normal Distribution N(μ, σ)
10.9.4 Central Moments of the Normal Distribution N(μ, σ)
10.9.5 Mean Deviation about the Mean of the Normal Distribution N(μ, σ)
10.9.6 Quartile Deviation of the Normal Distribution N(μ, σ)
10.9.7 Moment and Cumulant Generating Functions of N(μ, σ)
10.9.8 Additive Property of the Normal Distribution
10.9.9 Normal Distribution as Limiting Form of Binomial Distribution
10.9.10 Importance of Normal Distribution
Worked Examples 10(a)
Worked Examples 10(b)
Exercise10(a)
Exercise10(b)
Chapter 11 CORRELATION AND REGRESSION
11.1 Scatter Diagram
11.2 Correlation Coefficient
11.2.1 Property 1
11.2.2 Property 2
11.2.3 Property 3
11.2.4 Alternate Formula for r XY in Terms of Variances
11.3 Spearman’s Formula for the Rank Correlation Coefficient
11.4 Equation of the Regression Line of Y on X
11.4.1 Properties of Regression Coefficients
11.4.2 Standard Error of Estimate of Y
Worked Examples 11(a)
Worked Examples 11(b)
Exercise 11(a)
Exercise 11(b)
Chapter 12 MULTIPLE AND PARTIAL CORRELATIONS
12.1 A Note on Yule’s Subscript Notation
12.2 Plane of Regression
12.3 Properties of Residuals
12.4 Coefficient of Multiple Correlation
12.4.1 Multiple Correlation Coefficient in terms of Simple Correlation
Coefficients
12.5 Partial Correlation Coefficient in Terms of Simple Correlation Coefficients
Worked Examples 12
Exercise 12
Chapter 13 TESTS OF SIGNIFICANCE FOR LARGE SAMPLES
13.1 Sampling Distribution
13.2 Standard Errors
13.3 Tests of Significance
13.4 Critical Region and Level of Significance
13.5 One-tailed and Two-tailed Tests
13.6 Interval Estimation of Population Parameters
13.7 Assumptions used in Large Sample Tests
13.7.1 Test of Significance of the Difference between Sample Proportion and
Population Proportion
13.7.2 Test of Significance of the Difference between Two Sample Proportions
13.7.3 Test of Signification of the Difference between Sample Mean and
Population Mean
13.7.4 Test of Signification of the Difference between the Means of Two
Samples
13.7.5 Test of Significance of the Difference between the Sample S.D. and the
Population S.D
13.7.6 Test of Significance of the Difference between the S.D.’s of Two Large
Independent Samples
13.7.7 Test of Signification of the Difference between the Sample Correlation
Coefficient and the Population Correlation Coefficient
13.7.8 Test of Significance of the Difference between the Correlation Coefficient of Two Large Samples
Worked Examples 13
Exercise13
Chapter 14 EXACT SAMPLING DISTRIBUTIONS
14.1 Derivation of the Density Function of Chi-square (χ2)Distribution
14.1.1 Simple Moments and Cumulants of the χ2-Distribution
14.1.2 Mode and Skewness of the Chi-square Distribution
14.1.3 Limiting Form of Chi-Square Distribution for Large Values of n
14.1.4 Additive Property of Chi-square Distribution
14.1.5 Uses of χ2-Distribution
14.2 Derivation of the Density Function of Student’s t-Distribution
14.2.1 Moments of the Student’s t-Distribution and β1- and β2-Coefficients
14.2.2 Recurrence Relation for the Moments and Moment Generating Function
14.2.3 Mode of the t-Distribution
14.2.4 Limiting Form of t-Distribution for Large Values of n
14.2.5 Uses of t-Distribution
14.3 Snedecor’s F-Distribution – Definition and Derivation of P.D.F
14.3.1 Moments of F-Distribution
14.3.2 Recurrence Formula the Simple Moments and Value of the Variance
14.3.3 Mode of the F-Distribution and Form of F-Probability Curve
14.3.4 Uses of F-Distribution
14.3.5 Relations among t, F and χ2-Distributions
14.4 Definition and Derivation of the Density Function of Fisher’s Z-Distribution
14.4.1 Moments of Fisher’s Z-Distribution
14.4.2 Uses of Z-Distribution
Worked Examples 14
Exercise14
Chapter 15 TESTS OF SIGNIFICANCE FOR SMALL SAMPLES
15.1 Critical Values of t and the t-Table used in t-Tests
15.1.1 Test of Significance of the Difference between Sample Mean and
Population Mean
15.1.2 Test of Significance of the Difference between Means of Two Small Samples Drawn from the Same Normal Population
15.1.3 Test of Significance of Sample Correlation Coefficient
15.2 F-Test of Significance and F-Table
15.3 Chi-square (χ2) Tests
15.3.1 χ2-Test of Goodness of Fit
15.3.2 Conditions for the Validity of χ2-Test
15.3.3 χ2-Testof Independence of Attributes
Worked Examples 15(a)
Worked Examples 15(b)
Exercise15(a)
Exercise15(b)
Chapter 16 ESTIMATION THEORY
16.1 Interval Estimation
16.2 Point Estimation
16.2.1 Unbiased Estimator
16.2.2 Consistent Estimator
16.2.3 Efficient Estimator
16.2.4 Most Efficient Estimator
16.2.5 Sufficient Estimator
16.2.6 Neymann’s Factorisation Criterion for a Sufficient Estimator
16.3 Methods of Finding Estimators
16.3.1 Method of Maximum Likelihood
16.3.2 Method of Moments
Worked Examples 16(a)
Worked Examples 16(b)
Exercise 16(a)
Exercise16(b)
Chapter 17 TESTING OF HYPOTHESIS
17.1 Simple and Composite Hypotheses
17.2 Null Hypothesis and Alternative Hypothesis
17.3 Critical and Acceptance Regions
17.4 Errors in Hypothesis Testing and Power of a Test
17.5 Best Test, MP Test and UMP Test for a Simple Hypothesis
17.6 Neyman-Pearson Lemma
17.6.1 Likelihood Ratio Test (L.R.T.)
17.6.2 Two-tailed LRT for the Mean of a Normal Population
17.6.3 One-tailed Test for the Mean of a Normal Population
17.6.4 Two-tailed Test for the Variance of a Normal Population
17.6.5 One-tailed Test for the Variance of a Normal Population
17.7 Sequential Probability Ratio Test (SPRT)
17.8 Basic Assumption Associated with N.P. Tests
17.9.1 Sign Test
17.9.2 Wald’s Run Test
17.9.3 Rank Sum Test / Mann-Whitney-Wilcoxon Test
17.9.4 Median Test
17.9.5 Test for Randomness
Worked Examples 17(a)
Worked Examples 17(b)
Exercise17(a)
Exercise17(b)