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### Summary

### Summary

Tough Test Questions? Missed Lectures? Not Enough Time?

Fortunately, there's Schaum's. This all-in-one-package includes more than 750 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 20 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible.

More than 40 million students have trusted Schaum'sto help them succeed in the classroom and on exams.Schaum's is the key to faster learning and highergrades in every subject. Each Outline presents all theessential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.

This Schaum's Outline gives you

897 fully solved problems Concise explanations of all course fundamentals Information on conditional probability andindependence, random variables, binominal and normal distributions, sampling distributions, and analysis of varianceFully 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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### Author Notes

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.

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.

### Table of Contents

Part I Probability | p. 1 |

Chapter 1 Basic Probability | p. 3 |

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! | |

Chapter 2 Random Variables and Probability Distributions | p. 34 |

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 | |

Chapter 3 Mathematical Expectation | p. 75 |

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 | |

Chapter 4 Special Probability Distributions | p. 108 |

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 | |

Part II Statistics | p. 151 |

Chapter 5 Sampling Theory | p. 153 |

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 | |

Chapter 6 Estimation Theory | p. 195 |

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 | |

Chapter 7 Tests of Hypotheses and Significance | p. 213 |

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 | |

Chapter 8 Curve Fitting, Regression, and Correlation | p. 265 |

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 | |

Chapter 9 Analysis of Variance | p. 314 |

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 | |

Chapter 10 Nonparametric Tests | p. 348 |

Introduction | |

The Sign Test | |

The Mann-Whitney U Test | |

The Kruskal-Wallis H Test | |

The H Test Corrected for Ties | |

The Runs Test for Randomness | |

Further Applications of the Runs Test | |

Spearman's Rank Correlation | |

Chapter 11 Bayesian Methods | p. 372 |

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 | |

Appendix A Mathematical Topics | p. 411 |

Special Sums | |

Euler's Formulas | |

The Gamma Function | |

The Beta Function | |

Special Integrals | |

Appendix B Ordinates y of the Standard Normal Curve at z | p. 413 |

Appendix C Areas under the Standard Normal Curve from 0 to z | p. 414 |

Appendix D Percentile Values t p for Student's t Distribution with v Degrees of Freedom | p. 415 |

Appendix E Percentile Values x p 2 for the Chi-Square Distribution with v Degrees of Freedom | p. 416 |

Appendix F 95th and 99th Percentile Values for the F Distribution with v 1 v 2 Degrees of Freedom | p. 417 |

Appendix G Values of e -¿ | p. 419 |

Appendix H Random Numbers | p. 419 |

Subject Index | p. 420 |

Index for Solved Problems | p. 423 |