Syllabus

Probability spaces, conditional probability, independence; Random variables, distribution functions, probability mass and density functions, functions of random variables, standard univariate discrete and continuous distributions; Mathematical expectations, moments, moment generating functions, inequalities. Random vectors, joint, marginal and conditional distributions, conditional expectations, independence, covariance, correlation, standard multivariate distributions, functions of random vectors; Law of large numbers, central limit theorem. Sampling distributions; Point estimation – estimators, minimum variance unbiased estimation, maximum likelihood estimation, method of moments, consistency; Interval estimation; Testing of hypotheses – tests and critical regions, likelihood ratio tests; Linear regression.

• B. L. S. Prakasa Rao, A First Course in Probability and Statistics, World Scientific / Cambridge University Press India, 2009.
• R. V. Hogg, J. W. McKean, and A. Craig, Introduction to Mathematical Statistics, 6th Ed., Pearson Education India, 2006.

• Additional readings and instructor-provided material during the course.

Definition and sources of errors; Solutions of nonlinear equations – Bisection method, Newton's method and its variants, fixed point iterations, convergence analysis; Newton's method for non-linear systems. Finite differences and polynomial interpolation; Numerical integration – Trapezoidal and Simpson's rules, Gaussian quadrature. Initial value problems – Taylor series method, Euler and modified Euler methods, Runge-Kutta methods.

• D. Kincaid and W. Cheney, Numerical Mathematics and Computing, 7th Edition, Cengage, 2013.
• K. E. Atkinson, Introduction to Numerical Analysis, 2nd Edition, John Wiley, 1989.

• Additional lecture notes and material as provided by the instructor.

Linear systems – All variants of Gaussian elimination and LU factorization, Cholesky factorization. Linear least-squares problems – Normal equations, rotators and reflectors, QR factorization via rotators, reflectors and Gram-Schmidt orthonormalisation; QR method for linear least-squares problems; Rank-deficient least-squares problems. Singular Value Decomposition (SVD) – Numerical rank determination via SVD, solution of least squares problems, Moore-Penrose inverse, low-rank approximations using SVD, Principal Component Analysis, applications to data mining and image recognition. Eigenvalue Decomposition – Power, inverse power and Rayleigh quotient iterations, Schur decomposition, unitary similarity transformations of Hermitian matrices to tridiagonal form, QR algorithm, implementation of explicit QR algorithm for Hermitian matrices.

• L. N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
• D. S. Watkins, Fundamentals of Matrix Computation, 2nd Edition, Wiley, 2002.
• L. Elden, Matrix Methods in Data Mining and Pattern Recognition, SIAM, Philadelphia, 2007.

• Additional lecture notes and research articles as recommended by the instructor.

Review of fundamental data structures. Models of computation: Random access machines, space and time complexity measures, lower and upper bounds. Algorithm design techniques: Greedy method, divide-and-conquer, dynamic programming, and backtracking. Sorting and searching algorithms. Graph algorithms including traversal, shortest path, and spanning trees. Hashing techniques: Separate chaining, linear probing, and quadratic probing. Search trees: Binary search trees, AVL trees, and B-trees.

• T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd Edition, MIT Press, 2009.
• Jon Kleinberg and Éva Tardos, Algorithm Design, 1st Edition, Pearson Education, 2005.

• Supplementary lecture notes and instructor-recommended research papers.

Data Models: Overview of data models with emphasis on the relational model. Database Design: Conceptual design using the Entity-Relationship (E-R) model; mapping E-R models to relational schemas. Relational Algebra and Calculus: Formal query languages for relational databases. SQL: SQL queries, constraints, and triggers. Application Development: Stored procedures and database programming concepts.

• R. Ramakrishnan and J. Gehrke, Database Management Systems, 3rd Edition, McGraw Hill, 2003.

• Instructor-provided lecture notes and database project guides.

• T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 3rd Edition, MIT Press, 2009.
• Jon Kleinberg and Eva Tardos, Algorithm Design, 1st Edition, Pearson Education, 2006.
• R. Ramakrishnan and J. Gehrke, Database Management Systems, 3rd Edition, McGraw Hill, 2003.

• Additional materials and assignments provided by the instructor.

• Charles Dierbach, Introduction to Computer Science Using Python: A Computational Problem Solving Focus, John Wiley & Sons, 2012.

• Additional reading and programming exercises provided by the instructor.

• L. N. Trefethen and David Bau, Numerical Linear Algebra, SIAM, 1997.
• D. S. Watkins, Fundamentals of Matrix Computation, 2nd Edition, Wiley, 2002.
• D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Edition, AMS, 2002.
• K. E. Atkinson, Introduction to Numerical Analysis, 2nd Edition, John Wiley, 1989.

• Additional programming materials as suggested during the course.

• Sheldon M. Ross, Stochastic Processes, Wiley, 1995.
• W. R. Gilks, S. Richardson, and D. Spiegelhalter, Markov Chain Monte Carlo Methods in Practice, Chapman and Hall.

• Additional readings suggested during lectures.

• E. K. P. Chong and S. H. Zak, An Introduction to Optimization, 2nd Edition, Wiley India Pvt. Ltd., 2010.
• D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3rd Edition, Springer, 2010.

• Additional readings will be suggested during the course.

• E. Alpaydin, Introduction to Machine Learning, 3rd Edition, Prentice Hall (India), 2015.
• R. O. Duda, P. E. Hart and D. G. Stork, Pattern Classification, 2nd Edition, Wiley India, 2007.
• C. M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006.
• S. O. Haykin, Neural Networks and Learning Machines, 3rd Edition, Pearson Education (India), 2016.
• J. Shawe-Taylor and N. Cristianini, Kernel Methods for Pattern Analysis, Cambridge University Press, 2004.
• I. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT Press, 2017.
• R. Sutton, Reinforcement Learning – An Introduction, MIT Press, 1998.

• Relevant research papers in the area of Machine Learning.

• To be provided during the course.

• Additional materials and readings will be suggested during the course.

• G. James, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning: with Applications in R, Springer, 2013.
• W. John Braun and Duncan J. Murdoch, A First Course in Statistical Programming with R, Cambridge University Press, 2008.
• R Optimization Packages and Resources

• Additional readings and materials will be suggested during the course.

• Andy Kirk, Data Visualization: A Handbook for Data Driven Design, Sage Publications, 2016.
• Philipp K. Janert, Gnuplot in Action: Understanding Data with Graphs, Manning Publications, 2010.

• Additional readings and materials will be suggested during the course.