Probability Theory (1) - NU

1. Axiomatic Probability and CombinatoricsSample Spaces and Events: Defining random experiments, set theory operations (unions, intersections, complements), and Venn diagrams.Counting Principles: Utilizing permutations, combinations, and the multiplication rule to calculate classical sample space sizes.Probability Axioms: Applying Kolmogorov’s axioms to derive basic probability theorems and boundary rules.Conditional Probability: Evaluating dependent events using the conditional formula, the Law of Total Probability, and Bayes' Theorem for updating prior beliefs.2. Univariate Random VariablesDiscrete Random Variables: Constructing Probability Mass Functions (PMFs) and Cumulative Distribution Functions (CDFs).Continuous Random Variables: Defining Probability Density Functions (PDFs) and evaluating continuous probabilities using definite calculus integrals.Mathematical Expectation: Calculating the mean (Expected Value, \(\mu = E[X]\)), Variance (σ² = Var(X)), and Standard Deviation of a distribution.Moment Generating Functions (MGFs): Deriving \(M_X(t) = E[e^{tX}]\) as a structural tool to generate higher-order moments and uniquely identify distributions.3. Standard Discrete Probability DistributionsBinomial & Bernoulli: Modeling fixed success/failure counts across independent trials.Poisson: Counting occurrences over a continuous interval of time or space.Geometric & Negative Binomial: Measuring the number of trials required until a specified number of successes occur.Hypergeometric: Sampling without replacement from a finite population.4. Standard Continuous Probability DistributionsUniform: Modeling equally likely outcomes over a bounded interval \([a, b]\).Exponential: Measuring time-to-failure or waiting times, emphasizing its memoryless property.Normal (Gaussian): Utilizing the bell curve, Z-score transformations, and standard normal tables to evaluate symmetric physical variables.