Course Description
Core concepts of Probability (random variables/vectors and probability distributions) shall be covered in this course. At the end of the course, the students are expected to have enough familiarity with the subject to apply them in their own fields of study.
Course Content
- Basic definitions and ideas such as random experiment, sample space and event, Classical definition and relative frequency definition of probability, Axiomatic definition of probability, Elementary properties of probability function, Probability inequalities such as Boole’s inequality and Bonferroni inequality.
- Conditional probability and its basic properties, Examples of conditional probability and multiplication law, Theorem of total probability and related examples, Bayes theorem and related examples, Independent events.
- Random variables and their distribution function, Induced probability space, Discrete and continuous random variables, Function of random variables (Discrete and Continuous), Expectation and moments of random variables, MGF of random variables and its application, Markov, Chebyshev and Jensen’s inequality, Characteristics function and its application.
- Standard discrete distributions and their properties (e.g., Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson) Standard continuous distributions and their properties (e.g., Normal, Exponential, Gamma, Beta, Cauchy).
- Random vectors and their joint distribution functions, Marginal distribution, independent random variables, Conditional distribution of random vectors/variables, Expectation and moments of random vectors, Conditional Expectation, variance and covariance and their applications.
- Idea of limiting distribution, Convergence in distribution and probability, and related results, Convergence of moments and almost sure convergence, Various examples and counter examples.
- Weak law of large numbers, Central limit theorem, Applications, e.g., continuous mapping theorem and delta method.