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This text blends theory and applications, reinforcing concepts with practical real-world examples that illustrate the importance of probability to undergraduate students who will use it in their subsequent courses and careers. The author emphasizes the study of probability distributions that characterize random variables, because this knowledge is essential in performing parametric statistical analysis. Explanations include the "why" as well as the "how" of probability distributions for random variables to help engage students and further promote their understanding. In addition, the text includes a self-contained chapter on finite Markov chains, which introduces the basic aspects of Markov chains and illustrates their usefulness with several real examples.
- This text offers a clear and illustrative presentation of the fundamental aspects of probability, focusing on both its meaning and its usefulness.
- Throughout the text, topics are developed using a step-by-step approach and clarified by graphical illustrations.
- New concepts are illustrated and interpreted both graphically and analytically, addressing students' different learning styles and promoting their understanding by showing them that a problem can be solved in multiple ways.
- Boxes highlight important concepts, results, and the probability distributions, assisting students in study and review.
- The author places special emphasis on the development of the classical discrete and continuous probability distributions, their properties, and their interdisciplinary relevance to real life problems in our global society.
- The numerous exercises include items that test theoretical understanding as well as applications drawn from several areas in engineering, mathematics, and the sciences.
- A self-contained chapter on discrete finite Markov chains includes several applications that demonstrate their relevance as well as step-by-step development of the theory, clearly illustrated by state and tree diagrams.
- An extensive review of necessary mathematics that may not be covered in a basic calculus course—such as set theory, computational methods, binomial and multinomial theorems, matrices, Jacobians, and gamma and beta functions—is presented in an appendix for the convenience of the student.
Definition of Probability. Axiomatic Definition of Probability. Conditional Probability. Marginal Probabilities. Bayes' Theorem. Independent Events. Combinatorial Probability.
2. DISCRETE PROBABILITY DISTRIBUTIONS.
Discrete Probability Density Function. Cumulative Distribution Function. The Point Binomial Distribution. The Binomial Probability Distribution. The Poisson Probability Distribution. The Hypergeometric Probability Distribution. The Geometric Probability Distribution. The Negative Binomial Probability Distribution.
3. PROBABILITY DISTRIBUTIONS OF CONTINUOUS RANDOM VARIABLES.
Continuous Random Variable and Probability Density Function. Cumulative Distribution Function of a Continuous Random Variable. The Continuous Probability Distributions.
4. FUNCTIONS OF A RANDOM VARIABLE.
Introduction. Distribution of a Continuous Function of a Discrete Random Variable. Distribution of a Continuous Function of a Continuous Random Variable. Other Types of Derived Distributions.
5. EXPECTED VALUES, MOMENTS AND MOMENT GENERATING FUNCTIONS.
Mathematical Expectation. Properties of Expectation. Moments. Moment Generating Function.
6. TWO RANDOM VARIABLES.
Joint Probability Density Function. Bivariate Cumulative Distribution Function. Marginal Probability Distributions. Conditional Probability Density and Cumulative Distribution Functions. Independent Random Variables. Function of Two Random Variables. Expected Value and Moments. Conditional Expectation. Bivariate Normal Distribution.
7. SEQUENCE OF RANDOM VARIABLES.
Multivariate Probability Density Functions. Multivariate Cumulative Distribution Functions. Marginal Probability Distributions. Conditional Probability Density and Cumulative Distribution Functions. Sequence of Independent Random Variables. Functions of Random Variables. Expected Value and Moments. Conditional Expectation.
8. LIMIT THEOREMS.
Chebyshev's Inequality. Bernoulli's Law of Large Numbers. Weak and Strong Laws of Large Numbers. The Central Limit Theorem. The DeMoivre-Laplace Theorem. Normal Approximation to the Poisson Distribution. Normal Approximation to the Gamma Distribution.
9. FINITE MARKOV CHAINS.
Basic Concepts. N-Step Transitions Problems. Evaluation of Pn. Classification of States.