Theory and Problems of Probability and Statistics (Schaum's Outline Series) | ||

Spiegel (Murray) | ||

This Page provides (where held) the Abstract of the above Book and those of all the Papers contained in it. | ||

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**BOOK ABSTRACT: **__Preface__

- The important and fascinating subject of probability began in the 17th Century through efforts of such mathematicians as Fermat and Pascal to answer questions concerning games of chance. It was not until the 20th Century that a rigorous mathematical theory based on axioms, definitions and theorems was developed. As time progressed, probability theory found its way into many applications, not only in engineering, science and mathematics but in fields ranging from actuarial science, agriculture and business to medicine and psychology. In many instances the applications themselves contributed to the further development of the theory.
- The subject of statistics originated much earlier than probability and dealt mainly with the collection, organization and presentation of data in tables and charts. With the advent of probability it was realized that statistics could be used in drawing valid conclusions and making reasonable decisions on the basis of analysis of data, such as in sampling theory and prediction or forecasting.
- The purpose of this book is to present a modern introduction to probability and statistics using a background of calculus. For convenience the book is divided into two parts. The first deals with probability (and by itself can be used to provide an introduction to the subject), while the second deals with statistics.
- The book is designed to be used either as a textbook for a formal course in probability and statistics or as a comprehensive supplement to all current standard texts. It should also be of considerable value as a book of reference for research workers or to those interested in the field for self-study. The book can be used for a one-year course, or by a judicious choice of topics, a one-semester course.

→ M. R. Spiegel, September 1975

- Probability
- SETS AND PROBABILITY – 1

The Concept of a Set. Subsets. Universal Set and Empty Set. Venn Diagrams. Set Operations. Some Theorems Invoking Sets. Principle of Duality. Random Experiments. Sample Spaces. Events. The Concept of Probability. The Axioms of Probability. Some Important Theorems on Probability. Assignment of Probabilities. Conditional Probability. Theorems on Conditional Probability. Independent Events. Bayes’ Theorem or Rule. Combinatorial Analysis. Fundamental Principle of Counting. Tree Diagrams. Permutations. Combinations. Binomial Coefficients. Stirling’s Approximation to n!. - RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS – 38

Random Variables. Discrete Probability Distributions. Distribution Functions for Discrete Random Variables. Continuous Probability Distributions. Distribution Functions for Continuous Random Variables. Leibniz’s Rule. Graphical Interpretations. Joint Distributions. Independent Random Variables. Change of Variables. Probability Distributions of Functions of Random Variables. Convolutions. - MATHEMATICAL EXPECTATION – 76

Definition of Mathematical Expectation. Functions of Random Variables. Some Theorems on Expectation. The Variance and Standard Deviation. Some Theorems on Variance. Standardized Random Variables. Moments. Moment Generating Functions. Some Theorems on Moment Generating Functions. Characteristic Functions. Variance for Joint Distributions. Covariance. Correlation Coefficient. Conditional Expectation, Variance and Moments. Chebyshev’s Inequality. Law of Large Numbers. Other Measures of Central Tendency. Percentiles. Other Measures of Dispersion. Skewness and Kurtosis. - SPECIAL PROBABILITY DISTRIBUTIONS – 108

The Binomial or Bernoulli Distribution. Some Properties of the Binomial Distribution. The Law of Large Numbers for Bernoulli Trials. The Normal Distribution. Some Properties of the Normal Distribution. Relation Between Binomial and Normal Distributions. The Poisson Distribution. Some Properties of the Poisson Distribution. Relation Between the Binomial and Poisson Distributions. Relation Between the Poisson and Normal Distributions. The Central Limit Theorem. The Multinomial Distribution. The Hypergeometric Distribution. The Uniform Distribution. The Cauchy Distribution. The Gamma Distribution. The Beta Distribution. The Chi-Square Distribution. Student’s t Distribution. The F Distribution. Relationships Among Chi-Square, t and F Distributions. The Bivariate Normal Distribution. Miscellaneous Distributions.

- SETS AND PROBABILITY – 1
- Statistics
- SAMPLING THEORY – 155

Population and Sample. Statistical Inference. Sampling With and Without Replacement. Random Samples. Random Numbers. Population Parameters. Sample Statistics. Sampling Distributions. The Sample Mean. Sampling Distribution of Means. Sampling Distribution of Proportions. Sampling Distribution of Differences and Sums. The Sample Variance. Sampling Distribution of Variances. Case where Population Variance Is Unknown. Sampling Distribution of Ratios of Variances. Other Statistics. Frequency Distributions. Relative Frequency Distributions and Ogives. Computation of Mean, Variance and Moments for Grouped Data. - ESTIMATION THEORY – 194

Unbiased Estimates and Efficient Estimates. Point Estimates and Interval Estimates. Reliability. Confidence Interval Estimates of Population Parameters. Confidence Intervals for Means. Confidence Intervals for Proportions. Confidence Intervals for Differences and Sums. Confidence Intervals for Variances. Confidence Intervals for Variance Ratios. Maximum Likelihood Estimates. - TESTS OF HYPOTHESES AND SIGNIFICANCE – 211

Statistical Decisions. Statistical Hypotheses. Null Hypotheses. Tests of Hypotheses and Significance. Type I and Type II Errors. Level of Significance. Tests Involving the Normal Distribution. One-Tailed and Two-Tailed Tests. Special Tests of Significance for Large Samples. Special Tests of Significance for Small Samples. Relationship Between Estimation Theory and Hypothesis Testing. Operating Characteristic Curves. Power of a Test. Quality Control Charts. Fitting Theoretical Distributions to Sample Frequency Distributions. The Chi-Square Test for Goodness of Fit. Contingency Tables. Yates’ Correction for Continuity. Coefficient of Contingency. - CURVE FITTING, REGRESSION AND CORRELATION – 258

Curve Fitting. Regression. The Method of Least Squares. The Least-Squares Line. The Least-Squares Line in Terms of Sample Variances and Covariance. The Least-Squares Parabola. Multiple Regression. Standard Error of Estimate. The Linear Correlation Coefficient. Generalized Correlation Coefficient. Rank Correlation. Probability Interpretation of Regression. Probability Interpretation of Correlation. Sampling Theory of Regression. Sampling Theory of Correlation. Correlation and Dependence. - ANALYSIS OF VARIANCE – 306

The Purpose of Analysis of Variance. One-Way Classification or One-Factor Experiments. Total Variation. Variation Within Treatments. Variation Between Treatments. Shortcut Methods for Obtaining Variations. Linear Mathematical Model for Analysis of Variance. Expected Values of the Variations. Distributions of the Variations. The F Test for the Null Hypothesis of Equal Means. Analysis of Variance Tables. Modifications for Unequal Numbers of Observations. Two-Way Classification or Two-Factor Experiments. Notation for Two-Factor Experiments. Variations for Two-Factor Experiments. Analysis of Variance for Two-Factor Experiments. Two-Factor Experiments with Replication. Experimental Design.

- SAMPLING THEORY – 155

McGraw-Hill; Schaum's Outline Series in Mathematics, 1975

- Blue: Text by me; © Theo Todman, 2020
- Mauve: Text by correspondent(s) or other author(s); © the author(s)

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