# Schaum's Outline of Probability and Statistics, Fourth Edition

by: Murray R. Spiegel, John J. Schiller, R. Alu Srinivasan

**Abstract:**Practice problems with full explanations that reinforce knowledge, coverage of the most up-to-date developments in your course field, and in-depth review of practices and applications. Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!

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## Table of Contents

**A.**Preface to the Third Edition**B.**Preface to the Second Edition**C.**Preface to the First Edition**1.**Basic Probability**2.**Random Variables and Probability Distributions**3.**Mathematical Expectation**4.**Special Probability Distributions**5.**Sampling Theory**6.**Estimation Theory**7.**Tests of Hypotheses and Significance**8.**Curve Fitting, Regression, and Correlation**9.**Analysis of Variance**10.**Nonparametric Tests**11.**Bayesian Methods**A.**Mathematical Topics**B.**Ordinates y of the Standard Normal Curve at z**C.**Areas under the Standard Normal Curve from 0 to z**D.**Percentile Values tp for Student's t Distribution with ν Degrees of Freedom**E.**Percentile Values χ2p for the Chi-Square Distribution with ν Degrees of Freedom**F.**95th Percentile Values (0.05 Levels), F0.95, for the F Distribution**G.**Values of e-λ

## Tools & Media

## Expanded Table of Contents

**A.**Preface to the Third Edition**B.**Preface to the Second Edition**C.**Preface to the First Edition**1.**Basic Probability- 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 Coefficient
- Stirling's Approximation to n!
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**2.**Random Variables and Probability Distributions- Random Variables
- Discrete Probability Distributions
- Distribution Functions for Random Variables
- Distribution Functions for Discrete Random Variables
- Continuous Random Variables
- Graphical Interpretations
- Joint Distributions
- Independent Random Variables
- Change of Variables
- Probability Distributions of Functions of Random Variables
- Convolutions
- Conditional Distributions
- Applications to Geometric Probability
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**3.**Mathematical Expectation- 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
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**4.**Special Probability Distributions- The Binomial 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
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**5.**Sampling Theory- 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
- Computation of Mean, Variance, and Moments for Grouped Data
- Solved Problems
- Sampling distributions of differences and sums
- Sampling distribution of variances
- Supplementary Problems
- Answers to Supplementary Problems

**6.**Estimation Theory- 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 the Variance of a Normal Distribution
- Confidence Intervals for Variance Ratios
- Maximum Likelihood Estimates
- Solved Problems
- Confidence intervals for differences and sums
- Supplementary Problems
- Answers to Supplementary Problems

**7.**Tests of Hypotheses and Significance- 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
- P Value
- 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
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**8.**Curve Fitting, Regression, and Correlation- 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
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**9.**Analysis of Variance- 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
- Solved Problems
- Supplementary Problems
- Answers to Supplementary Problems

**10.**Nonparametric Tests**11.**Bayesian Methods- Subjective Probability
- Prior and Posterior Distributions
- Sampling From a Binomial Population
- Sampling From a Poisson Population
- Sampling From a Normal Population with Known Variance
- Improper Prior Distributions
- Conjugate Prior Distributions
- Bayesian Point Estimation
- Bayesian Interval Estimation
- Bayesian Hypothesis Tests
- Bayes Factors
- Bayesian Predictive Distributions
- Solved Problems
- Supplementary Problems

**A.**Mathematical Topics**B.**Ordinates y of the Standard Normal Curve at z**C.**Areas under the Standard Normal Curve from 0 to z**D.**Percentile Values tp for Student's t Distribution with ν Degrees of Freedom**E.**Percentile Values χ2p for the Chi-Square Distribution with ν Degrees of Freedom**F.**95th Percentile Values (0.05 Levels), F0.95, for the F Distribution**G.**Values of e-λ

**Book Details**

**Title: **Schaum's Outline of Probability and Statistics, Fourth Edition

**Publisher: **McGraw-Hill: New York, Chicago, San Francisco, Lisbon, London, Madrid, Mexico City, Milan, New Delhi, San Juan, Seoul, Singapore, Sydney, Toronto

**Copyright / Pub. Date: **2013 The McGraw-Hill Companies, Inc.

**ISBN: **9780071795579

**Authors:****Murray R. Spiegel**
(deceased) received the M.S. degree in physics and the Ph.D. in mathematics from Cornell University. He had positions at Harvard University, Columbia University, Oak Ridge and Rensselaer Polytechnic Institute, and served as a mathematical consultant at several large companies. His last position was professor and chairman of Mathematics at the Rensselaer Polytechnic Institute, Hartford Graduate Center.**John J. Schiller**
is an associate professor of mathematics at Temple University. He received his Ph.D. at the University of Pennsylvania.**R. Alu Srinivasan**
is a professor of mathematics at Temple University. He received his Ph.D. at Wayne State University and has published extensively in probability and statistics.

**Description: **
Practice problems with full explanations that reinforce knowledge, coverage of the most up-to-date developments in your course field, and in-depth review of practices and applications. Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!