The axiomatic definition of experiment and probability. Conditional Probability. Bayes’ Theorem, Notion of independence. Repeated trials. Bernoulli trials and their limiting forms. The concept of a random variable. Probability distribution and density functions. Probability mass functions. Examples of random variables: Normal(Gaussian), Poisson, Gamma, Exponential, Laplace, Cauchy, Rayleigh, etc. Bayes’ Theorem revisited. Functions of one random variable and their density functions. Expected value of a random variable: Mean, Variance, Moments and Characteristic functions. Two random variables: Joint distribution and joint density functions of two random variables, Independence. One function of two random variables. Two functions of two random variables. Order statistics. Joint moments, Uncorrelatedness, Orthogonality, Joint characteristic function. Jointly Gaussian random variables. Conditional distribution and conditional expected values. The central limit theorem. The principle of maximum likelihood. Elements of parameter estimation. Maximum likelihood estimation for unknown parameters. Unbiased estimators and their variances. Co-listed as BE 6453.
Prerequisites: Graduate status and MA 3012. *Online version available.