JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

Pre-requisites

Mathematics courses of first year of study.

Course Objectives

To learn

• The ideas of probability and random variables and various discrete and continuous probability distributions and their properties.
• The basic ideas of statistics including measures of central tendency, correlation and regression.
• The statistical methods of studying data samples.
• Differentiation and integration of complex valued functions.
• Evaluation of integrals using Cauchy’s integral formula and Cauchy’s residue theorem.
• Expansion of complex functions using Taylor’s and Laurent’s series. Course outcomes: After learning the contents of this paper the student must be able to
• Formulate and solve problems involving random variables and apply statistical methods for analyzing experimental data.
• Apply concept of estimation and testing of hypothesis to case studies.
• Analyze the complex function with reference to their analyticity, integration using Cauchy’s integral and residue theorems.
• Taylor’s and Laurent’s series expansions of complex function.

UNIT - I: Basic Probability

Probability spaces, conditional probability, independent events, and Baye’s theorem. Random variables: Discrete and continuous random variables, Expectation of Random Variables, Variance of random variables

UNIT - II: Probability distributions

Binomial, Poisson, evaluation of statistical parameters for these distributions, Poisson approximation to the binomial distribution, Continuous random variables and their properties, distribution functions and density functions, Normal and exponential, evaluation of statistical parameters for these distributions

UNIT - III: Estimation & Tests of Hypotheses

Introduction, Statistical Inference, Classical Methods of Estimation.: Estimating the Mean, Standard Error of a Point Estimate, Prediction Intervals, Estimating a Proportion for single sample, Difference between Two Means, difference between two proportions for two Samples. Statistical Hypotheses: General Concepts, Testing a Statistical Hypothesis, Tests Concerning a Single Mean, Tests on Two Means, Test on a Single Proportion, Two Samples: Tests on Two Proportions.

UNIT - IV: Complex Differentiation

Limit, Continuity and Differentiation of Complex functions, Analyticity, Cauchy-Riemann equations (without proof), finding harmonic conjugate, elementary analytic functions (exponential, trigonometric, logarithm) and their properties, Conformal mappings, Mobius transformations.

UNIT - V: Complex Integration

Line integral, Cauchy’s theorem, Cauchy’s Integral formula, Zeros of analytic functions, Singularities, Taylor’s series, Laurent’s series, Residues, Cauchy Residue theorem (All theorems without Proof).

TEXT BOOKS:

• 1. B.S. Grewal, Higher Engineering Mathematics, Khanna Publishers, 35th Edition, 2010.
• 2. Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, keying Ye, Probability and statistics for engineers and scientists, 9th Edition, Pearson Publications.

REFERENCE BOOKS:

• 1. Fundamentals of Mathematical Statistics, Khanna Publications, S. C. Guptha and V. K. Kapoor.
• 2. Miller and Freund’s, Probability and Statistics for Engineers, 8th Edition, Pearson Educations.
• 3. N.P. Bali and Manish Goyal, A text book of Engineering Mathematics, Laxmi Publications, Reprint, 2010.
• 4. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Edition, Mc-Graw Hill, 2004.