Mathematical Foundation of AI, ML and Data Science (Elective-I)

Unit I

Descriptive Statistics & Probability: Diagrammatic representation of data, measures of central tendency, dispersion, skewness and kurtosis, correlation, inference procedure for correlation coefficient, bivariate and multiple correlation, linear regression and its inference procedure, multiple regression. Probability: conditional probability, independent events, Bayes' theorem, random variables, discrete and continuous probability distributions, expectation and variance, Markov inequality, Chebyshev's inequality, central limit theorem.

Unit II

Inferential Statistics & Regression Analysis: Sampling & Confidence Interval, Inference & Significance, Estimation and Hypothesis Testing, Goodness of fit, Test of Independence, Permutations and Randomization Test, t-test/z-test (one sample, independent, paired), ANOVA, chi-square. Linear Methods for Regression: multiple regression analysis, QR orthogonalization by Householder transformations, singular value decomposition (SVD), linear dimension reduction using PCA.

Unit III

Random Numbers & Monte Carlo: Random number generation, inverse-transform, acceptance-rejection, transformations, multivariate probability calculations. Monte Carlo Integration: simulation, variance reduction, Monte Carlo hypothesis testing, antithetic variables/control variates, importance sampling, stratified sampling. Markov chain Monte Carlo (MCMC): Markov chains, Metropolis-Hastings algorithm, Gibbs sampling, convergence.

Unit IV

Vector Spaces & Inner Product Spaces: Vector Space, Subspace, Linear Combination, Linear Independence, Basis, Dimension, Finding a Basis, Coordinates, Change of Basis. Inner Product Spaces: Inner Product, Length, Orthogonal Vectors, Triangle Inequality, Cauchy-Schwarz Inequality, Orthonormal Basis, Gram-Schmidt Process.

Unit V

Linear Transformations & Eigenvalues: Linear Transformations and Matrices, Kernel and Range, Change of Basis. Eigenvalues and Eigenvectors: Definition, Diagonalization, Symmetric Matrices and Orthogonal Diagonalization.