Mathematical Foundation AI, ML and Data Science

Diagrammatic representation of data, measures of central tendency, measures of dispersion, measures of skewness and kurtosis, correlation, inference procedure for correlation coefficient, bivariate correlation, multiple correlations, linear regression and its inference procedure, multiple regression. Probability: Measures of probability, conditional probability, independent event, Bayes’ theorem, random variable, discrete and continuous probability distributions, expectation and variance, markov inequality, chebyshev’s inequality, central limit theorem.

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

Random number generation, Inverse-transform, acceptance-rejection, transformations, multivariate probability calculations. Monte Carlo Integration: Simulation and Monte Carlo integration, 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

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

Linear Transformations- Linear Transformations and Matrices for Linear Transformation, Kernel and Range of a Linear Transformations, Change of Basis Eigenvalues and Eigenvectors- Definition of Eigenvalue and Eigenvector, Diagonalization, Symmetric Matrices and Orthogonal Diagonalization