Probability for Data Science

At the end of the course the students should be able to: 1. analyse and interpret real-world statistical events; 2. utilise various principles and concepts from the broad theory of probability and adjoining statistical and mathematical fields; 3. apply statistical principles and concepts to analyse data; and 4. analyse and interpret real-world statistical events by applying various principles and concepts from the broad theory of probability and adjoining statistical and mathematical fields.

Experiments, sample spaces, outcomes and events. Generation of Statistical events from set theory (Venn diagrams). Concepts and principles of Probability (probability axioms). Random variables. The Law of Total Probability, Bayes’ Theorem, Independence. Permutation and Combination. Introduction to Probability and distribution functions. The probability density function. Basic distributions: Bernoulli Trials, Binomial, Hyper geometric, Poisson, and Normal. Exploratory data analysis. Combinatorial analysis. Probability models for the study of random phenomena in finite sample generating functions and its properties. Chebyshev’s inequality and limit theorems in probability. Central limit theorem. Bivariate, marginal and conditional distributions. Variance and covariance. Probability mass function. Geometric distribution. Sampling with and without replacement. Hypergeometric distribution. Bounding probabilities, tail sum formula. Markov’s inequality. The exponential distribution, moments, memoryless property, hazard function. Definition of a Markov chain and probability transition matrices. Equilibrium behaviour of Markov chains: computer demonstration and ergodic, limiting and stationary interpretations. Mean and variance of linear combination of two random variables. The joint Moment generating function (MGF) and MGF of the sum. Definition of absorbing Markov chains, structural results, hitting probabilities and expected hitting times.