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Full Description
This text focuses on the utility of probability in solving real-world problems for students in a one-semester calculus-based probability course. Theory is developed to a practical degree and grounded in discussion of its practical uses in solving real-world problems. Numerous applications using up-to-date real data in engineering and the life, social, and physical sciences illustrate and motivate the many ways probability affects our lives. The text's accessible presentation carefully progresses from routine to more difficult problems to suit students of different backgrounds, and carefully explains how and where to apply methods. Students going on to more advanced courses in probability and statistics will gain a solid background in fundamental concepts and theory, while students who must apply probability to their courses engineering and the sciences will develop a working knowledge of the subject and appreciation of its practical power.
Contents
1. PROBABILITY IN THE WORLD AROUND US.
Why Study Probability? Deterministic and Probabilistic Models. Modeling Reality. Deterministic Models. Probabilistic Models. Applications in Probability. A Brief Historical Note. A Look Ahead.
2. FOUNDATIONS OF PROBABILITY.
Understanding Randomness: An Intuitive Notion of Probability. Randomness with Known Structure. Randomness with Unknown Structure. Sampling a Finite Universe. Sample Space and Events. Definition of Probability. Counting Rules Useful in Probability. More Counting Rules Useful in Probability. Summary.
3. CONDITIONAL PROBABILITY AND INDEPENDENCE.
Conditional Probability. Independence. Theorem of Total Probability and Bayes' Rule. Odds, Odds Ratios, and Relative Risk. Summary.
4. DISCRETE PROBABILITY DISTRIBUTIONS.
Random Variables and Their Probability Distributions. Expected Values of Random Variables. The Bernoulli Distribution. The Binomial Distribution. Probability Function. Mean and Variance. History and Applications. The Geometric Distribution. Probability Function. Mean and Variance. An Alternate Parameterization: Number of Trials Versus Number of Failures. The Negative Binomial Distribution. Probability Function. Mean and Variance. An Alternate Parameterization: Number of Trials Versus Number of Failures. History and Applications. The Poisson Distribution. Probability Function. Mean and Variance. History and Applications. The Hypergeometric Distribution. The Probability Function. Mean and Variance. History and Applications. The Moment-generating Function. The Probability-generating Function. Markov Chains. Summary.
5. CONTINUOUS PROBABILITY DISTRIBUTIONS.
Continuous Random Variables and Their Probability Distributions. Expected Values of Continuous Random Variables. The Uniform Distribution. Probability Density Function. Mean and Variance. History and Applications. The Exponential Distribution. Probability Density Function. Mean and Variance. Properties. History and Applications. The Gamma Distribution. Probability Density Function. Mean and Variance. History and Applications. The Normal Distribution. The Normal Probability Density Function. Mean and Variance. Calculating Normal Probabilities. Applications to Real Data. Quantile-Quantile (Q-Q) Plots. History. The Beta Distribution. Probability Density Function. Mean and Variance. H istory and Applications. The Weibull Distribution. Probability Density Function. Mean and Variance. History and Applications to Real Data. Reliability. Hazard Rate Function. Series and Parallel Systems. Redundancy. Moment-generating Functions for Continuous Random Variables. Expectations of Discontinuous Functions and Mixed Probability Distributions. Summary.
6. MULTIVARIATE PROBABILITY DISTRIBUTIONS.
Bivariate and Marginal Probability Distributions. Conditional Probability Distributions. Independent Random Variables. Expected Values of Functions of Random Variables. Conditional Expectations. The Multinomial Distribution. More on the Moment-Generating Function. Compounding and Its Applications. Summary.
7. FUNCTIONS OF RANDOM VARIABLES.
Introduction. Functions of Discrete Random Variables. Method of Distribution Functions. Method of Transformations in One Dimension. Method of Conditioning. Method of Moment-Generating Functions. Gamma Case. Normal Case. Normal and Gamma Relationships. Method of Transformation—Two Dimensions. Order Statistics. Probability-Generating Functions: Applications to Random Sums of Random Variables. Summary.
8. SOME APPROXIMATIONS TO PROBABILITY DISTRIBUTIONS: LIMIT THEOREMS.
Introduction. Convergence in Probability. Convergence in Distribution. The Central Limit Theorem. Combination of Convergence in Probability and Convergence in Distribution. Summary.
9. EXTENSIONS OF PROBABILITY THEORY.
The Poisson Process. Birth and Death Processes: Biological Applications. Queues: Engineering Applications. Arrival Times for the Poisson Process. Infinite Server Queue. Renewal Theory: Reliability Applications. Summary.
