R18 B.TECH MECHANICAL ENGG.

MA301BS: PROBABILITY AND STATISTICS & COMPLEX VARIABLES

B.Tech. II Year I Sem. L T/P/D C

3 1/0/0 4

Pre-requisites: Mathematical Knowledge at pre-university level

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 analysing experimental data.
• Analyse 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 8 L

Probability spaces, conditional probability, independent events, and Bayes’ theorem.

Random variables: Discrete and continuous random variables, Expectation of Random Variables, Moments, Variance of random variables

UNIT - II: Probability distributions 10 L

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: Testing of Hypothesis 10 L

Test of significance: Basic of testing of Hypothesis. Null and alternate Hypothesis, types of errors, level of significance, critical region.

Large sample test for single proportion, difference of proportions, single mean, difference of means; small sample tests: Test for single mean, difference of means and test for ratio of variances

UNIT - IV: Complex Variables (Differentiation) 10 L

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.

UNIT - V: Complex Variables (Integration) 10 L

Line integral, Cauchy’s theorem, Cauchy’s Integral formula, Zeros of analytic functions, Singularities, Taylor’s series, Laurent’s series; Residues, Cauchy Residue theorem, Conformal mappings, Mobius transformations and their properties.

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.
3. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th Ed., Mc-Graw Hill, 2004.

REFERENCES:

1. Fundamentals of Mathematical Statistics, Khanna Publications, S. C. Gupta and V. K. Kapoor.
2. Miller and Freund’s, Probability and Statistics for Engineers, 8th Edition, Pearson Educations
3. S. Ross, A First Course in Probability, 6th Ed., Pearson Education India, 2002.
4. Erwin kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley & Sons, 2006.
5. N.P. Bali and Manish Goyal, A text book of Engineering Mathematics, Laxmi Publications, Reprint, 2010.