Probability Theory Lecture 9

What is probability, Why do we need a mathematical theory, Basic Principles of Probability, Sample space, Events, Probability measure, Probabilistic modelling, Conditional probability, Independent events, Random variables, Expectationanddistributions, Independence and conditioning, Bernoulli Processes, Counting successes and binomial distribution, Arrival times and geometric distribution, The law of large numbers, From discrete to continuous arrivals, Continuous Random Variables, Expectation and integrals, Joint and conditional densities, Independence, Lifetimes and Reliability, Lifetimes, Minima and maxima, Reliability, A random process, Poisson Processes, Counting processes and Poisson processes, Superposition and thinning, Nonhomogeneous Poisson processes, Random Walks, What is a random walk, Hitting times, Gambler’s ruin, Biased random walks, Brownian Motion, The continuous time limit of a random walk, Brownian motion and Gaussian distribution, The central limit theorem, Jointly Gaussian variables, Sample paths of Brownian motion, Branching Processes, The Galton-Watson process, Extinction probability, Markov Chains, Markov chains and transition probabilities, Classification of states, First step analysis, Steady-state behaviour, The law of large numbers revisited.