Probability and Probability Distributions With R Applications

This textbook is designed to serve as both a primary and supplementary resource for all academic departments offering a course in probability. Each chapter is systematically structured to begin with a theoretical exposition, followed by detailed, fully worked examples. To support deeper understanding and practical application of probability concepts, each section includes corresponding R programming implementations. The content encompasses the foundational concepts of set theory, construction of sample spaces, conditional probability, independence, and Bayes' theorem. It further explores the concept of random variables, properties of discrete and continuous random variables, probability mass functions, probability density functions, and cumulative distribution functions. Scenarios involving univariate, bivariate, and multivariate random variables are thoroughly analyzed. Additionally, the text covers independence of random variables, conditional probability functions, quantiles, and key statistical measures associated with probability distributions including expected value, variance, moments, moment-generating functions, covariance, correlation, characteristic functions, and factorial moment-generating functions. Essential inequalities such as Markov, Chebyshev, and Cauchy–Schwarz, along with the Central Limit Theorem, are presented with comprehensive exercises and R-based solutions. The book also provides an in-depth examination of commonly used discrete and continuous probability distributions, including: Bernoulli, Binomial, Multinomial, Geometric, Negative Binomial, Hypergeometric, Generalized Hypergeometric, Poisson, Discrete Uniform, Continuous Uniform, Normal, Standard Normal, Bivariate Normal, Log-Normal, Exponential, Gamma, Beta, and Cauchy distributions. For each distribution, the probability and distribution functions, distributional shapes, expected values, moments, and moment-generating functions are derived and illustrated with examples and R programming applications.