Fundamentals of mathematical statistics 8170147913

Table of contents :
Title Page
Preface
CONTENTS
1. Introduction - Meaning and Scope
1.1 Origin and Development of Statitics
1.2 Definition or Statistics
1.3 Importance and Scope of Statistics
1.4 Limitations or Statistics
2. Frequency Distributions and Mesures of Central Tendency
2.1 Frequency Distributions
2.2 Graphic Representation or Frequency Distribution
2.3 Averages or Measures of Central Tendency or Measures of Location
2.4 Requisites for an Ideal Measure of Central Tendency
2.5 Arithmetic Mean
2.6 Median
2.7 Mode
2.8 Geometric Mean
2.9 Harmonic Mean
2.10 Selection of an Average
2.11 Partition Values
3. Measures of Dispersion, Skewness and Kurtosis
3.1 Dispersion
3.2 Characteristics of an Ideal Measure of Dispersion
3.3 Measures of Dispersion
3.4 Range
3.5 Quartitle Deviation
3.6 Mean Deviation
3.7. Standard Deviation and Root Mean Square Deviation
3.8. Coefficient of Dispersion
3.9 Moments
3.10. Pearson's b and g Coefficients
3.11 Factorial Moments
3.12 Absolute Moments
3.13 Skewness
3.14 Kurtosis
4. Theory of Probability
4.1 Introduction
4.2 Short History
4.3 Definitions of Various Terms
4.4 Mathematical Tools: Preliminary Notions of Sets
4.5 Axiomatic Approach to Probability
4.6 Probability - Mathematical Notion
4.7 Multiplication Law of Probability and Conditional Probability
4.8 Bayes Theorem
4.9 Geometric probability
5. Random Variables-Distribution Functions
5.1 Random Variable
5.2 Distribution Function
5.3 Discrete Random Variable
5.4 Continuous Random Variable
5.5 Joint Probability Law
5.6 Transformation of One-dimensional Random Variable
5.7 Transformation of Two-dimensional Random Variable
6. Mathematical Expectation, Generating Functions and Law of Large Numbers
6.1 Mathematical Expectation
6.2 Expectation of a Function of a Random Variable
6.3 Addition Theorem of Expectation
6.4 Multiplication Theorem of Expectation
6.5 Expectation of a Linear Combiriation of Random Variables
6.6 Covariance
6.7 Variance of a Linear Combination of Random Variables
6.8 Moments of Bivariate Probability Distributions
6.9 Conditional Expectation and Conditional Variance
6.10 Moment Generating Function
6.11 Cumulants
6.12 Characteristic Function
6.13 Chebychev's Inequality
6.14 Convergence in probability
6.15 Weak Law of Large Numbers
6.16 Borel-Cantelli Lemma
6.17 Probability Generating Function
7. Theoretical Discrete Probability Distributions
7.0 Introduction
7.1 Bernoulli Distribution
7.2 Binomial Distribution
7.3 Poisson Distribution
7.4 Negative Binomial Distribution
7.5 Geometric: Distribution
7.6 Hypeometric Distribution
7.7 Multinomial Distribution
7.8 Discrete Uniform Distribution
7.9 Power Series Distribution
8. Theoretical Continuous Distributions
8.1 Rectangular (or Uniform Distribution)
8.2 Normal Distribution
8.3 Gamma Distribution
8.4 Beta Distribution of First Kind
8.5 Bela Distribution of Second Kind
8.6 The Exponential Distribution
8.7 Laplace (Double Exponential) Distribution
8.8. Weibul Distribution
8.9 Cauchy's Distribution
8.10 Central Limit Theorem
8.11 Compound distributions
8.12 Pearson's Distributions
8.13 Variate Tranformations
8.14 Order Statistics
8.15 Truncated Distributions
9. Curve Fitting and Principle of Least Squares
9.1 Curve Fitting
9.2 Most Plausible Solution of a System of Linear Equations
9.3 Conversion of Data to Linear Form
9.4 Selection of Type of Curve be Fitted
9.5 Curve Fitting by Orthogonal Polynomials
10. Correlation and Regression
10.1 Bivariate Distribution, Correlation
10.2 Scatter Diagram
10.3 Karl Pearson Coefficient or Correlation
10.4 Calculation of tbe Correlation Coefficient (or a Bivariate Frequency Distribution)
10.5 Probable Error of Correlation Coefficient
10.6 Rank Correlation
10.7 Regression
10.8 Correlation Ratio
10.9 Intra-class Correlation
10.10 Bivariate Normal Distribution
10.11 Multiple and Partial Correlation
10.12 Plane of Regression
10.13 Properties or Residuals
10.14 Coefficient of Multiple Correlation
10.15 Coefficient of Partial Correlation
10.16 Multiple Correlation in Terms or Total and Partial Correlations
10.17 Expression for Regression Coefficients in Terms of Regression Coefficients of Lower Order
10.18 Expression for Partial Correlation Coefficient in Terms of Correlation Coefficients of Lower Order
11. Theory of Attributes
11.1 Introduction
11.2 Notations
11.3 Dichotomy
11.4 Classes and Class Frequencies
11.5 Class Symbols as Operators
11.6 Consistenc of Data
11.7 Independence of Attributes
11.8 Associatio of Attributes
12. Sampling and Large Sample Tests
12.1 Sampling-Introduction
12.2 Types or Sampling
12.3 Parameter and Statistic
12.4 Tests of Significance
12.5 Null Hypothesis
12.6 Errors in Sampling
12.7 Critical Region and Level or Significance
12.8 Test of Significance for Large Samples
12.9 Sampling for Attributes
12.10 Sampling of Variables
12.11 Unbiased Estimate for population Mean and Variance
12.12 Standard Error of Sample Mean
12.13 Test of Significance for Single Mean
12.14 Test of Significance for Difference of Means
12.15 Test of Significance for the Difference of Standard Deviations
13. Exact Sampling Distributions (Chi-square Distribution)
13.1 Chi-Square Variate
13.2 Derivation or the Chi-square Distribution.First Method-Method of Moment Generating Function
13.3 M.G.F. of X2-distribution
13.4 Chi-square Probability Curve
13.5 Conditions for tbe Validity of X^2 test
13.6 Linear Transformatlon
13.7 Applications or Chi-square Distribution
13.8 Yates' Correction
13.9 Brandt and Snedecor Formula for 2 x k Contingency Table
13.10 Bartlett's Test for Homogeneity of Several Independent Estimates of the Same Population Variance
13.11 X2-Test for Pooling the Probabilities from Independent Tests to give a Single Test of Significance
13.12 Non-central X2-distribution
14. Exact Sampling Distributions (t, F AND Z DISTRIBUTIONS)
14.1 Introduction
14.2 Student's 't'
14.3 Distribution of Sample Correlation Coefficient when Population Correlation Coefficient rho = 0
14.4 Non-central t-distribution
14.5 F-statistic. Definition
14.6 Non-Central F-dlstribution
14.7 Fisher's z-distribution
14.8 Fisher's z-transformation
15. Statistical Inference I (Theory of Estimation)
15.1 Introduction
15.2 Characteristics of Estimators
15.3 Consistency
15.4 Unbiasedness
15.5 Efficient Estimators
15.6 Sufficiency
15.7 Cramer-Rao Inequality
15.8 Complete Family or Distributions
15.9 MVU and Blackwellisation
15.10 Methods of Estimation
15.11 Method of Maximum Likelihood Estimation
15.12 Method of Minimum Variance
15.13 Method ot Moments
15.14 Method of Least Squares
15.15 Confidence Interval and Confidence Limits
16. Statistical Infernce II (Testing of Hypothesis, Non-parametric Methods and Sequential Analysis J
16.1 Introduction
16.2 Statistical hypothesis-Simple and Composite
16.3 Steps in Solvlog Testiog of Hypothesis Problem
16.4 Optimum Test Under Different Situations
16.5 Neyman J. and Pearson, E.S. Lemma
16.6 Likelihood Ratio Test
16.7 How the Likelihood Ratio Criterion can be used to Obtain Various Standard tests of Significance
16.8 Non-parametric Methods
16.9 Sequential Analysis
App.: Numerical Tables
INDEX

Citation preview

FUNDAMENTALS OF MATHEMATICAL STATISTICS (A Modern Approach) A Textbook written completely on modern lines for Degree, Honours, Post-graduate Students of al/ Indian Universities and ~ndian Civil Services, Indian Statistical Service Examinations.

(Contains, besides complete theory, more than 650 fully solved examples and more than 1,500 thought-provoking Problems with Answers, and Objective Type Questions)

V.K. KAPOOR

S.C. GUPTA Reader in Statistics Hindu College, University of Delhi Delhi

Reader in Mathematics Shri Ram Coffege of Commerce University of Delhi Delhi

Tenth Revised Edition (Greatly Improved) '.

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