Probability and complex function

MA3303

PROBABILITY AND COMPLEX FUNCTIONS

OBJECTIVES

• This course aims at providing the required skill to apply the statistical tools in engineering problems.

• To introduce the basic concepts of probability and random variables.

• To introduce the basic concepts of two dimensional random variables.

• To develop an understanding of the standard techniques of complex variable theory in particular analytic function and its mapping property.

• To familiarize the students with complex integration techniques and contour integration techniques which can be used in real integrals.

• To acquaint the students with Differential Equations which are significantly used in engineering problems.

UNIT I

PROBABILITY AND RANDOM VARIABLES

Axioms of probability - Conditional probability - Baye's theorem - Discrete and continuous random variables - Moments - Moment generating functions - Binomial, Poisson, Geometric, Uniform, Exponential and Normal distributions - Functions of a random variable.

UNIT II

TWO-DIMENSIONAL RANDOM VARIABLES

Joint distributions Marginal and conditional distributions - Covariance Correlation and linear regression Transformation of random variables - Central limit theorem (for independent and identically distributed random variables).

UNIT III

ANALYTIC FUNCTIONS

Analytic functions - Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates Properties - Harmonic conjugates - Construction of analytic function - Conformal mapping Mapping by functions WZ+C, cz,- - Bilinear transformation.

UNIT IV

COMPLEX INTEGRATION

Line integral - Cauchy's integral theorem - Cauchy's integral formula - Taylor's and Laurent's series - Singularities - Residues - Residue theorem - Application of residue theorem for evaluation of real integrals - Applications of circular contour and semicircular contour (with poles NOT on real axis).

UNIT V

ORDINARY DIFFERENTIAL EQUATIONS KNOWLEDGE

Higher order linear differential equations with constant coefficients - Method of variation of parameters - Homogenous equation of Euler's and Legendre's type - System of simultaneous linear first order differential equations with constant coefficients - Method of undetermined coefficients.

TOTAL: 60 PERIODS

COURSE OUTCOMES:

Upon successful completion of the course, students will be able to:

CO1: Understand the fundamental knowledge of the concepts of probability and have knowledge of standard distributions which can describe real life phenomenon.

CO2: Understand the basic concepts of one and two dimensional random variables and apply in engineering applications.

CO3: To develop an understanding of the standard techniques of complex variable theory in particular analytic function and its mapping property.

CO4: To familiarize the students with complex integration techniques and contour integration techniques which can be used in real integrals.

CO5: To acquaint the students with Differential Equations which are significantly used in engineering problems.

TEXT BOOKS

1. Johnson. R.A., Miller. I and Freund. J., "Miller and Freund's Probability and Statistics for Engineers", Pearson Education, Asia, 9th Edition, 2016.

2. Milton. J. S. and Arnold. J.C., "Introduction to Probability and Statistics", Tata McGraw Hill, 4th Edition, 2007.

3. Grewal.B.S., "Higher Engineering Mathematics", Khanna Publishers, New Delhi, 44th Edition, 2018.

REFERENCES

1. Devore. J.L., "Probability and Statistics for Engineering and the Sciences", Cengage Learning, New Delhi, 8th Edition, 2014.

2. Papoulis. A. and Unnikrishnapillai S., "Probability, Random Variables and Stochastic Processes", McGraw Hill Education India, 4th Edition, New Delhi, 2010.

3. Ross S.M., "Introduction to Probability and Statistics for Engineers and Scientists", 5th Edition, Elsevier, 2014.

4. Spiegel. M.R., Schiller. J. and Srinivasan. R.A., "Schaum's Outline of Theory and Problems of Probability and Statistics", Tata McGraw Hill Edition, 4th Edition, 2012.

5. Walpole. R.E., Myers. R.H., Myers. S.L. and Ye. K., "Probability and Statistics for Engineers and Scientists", Pearson Education, Asia, 9th Edition, 2010.

6. Kreyszig.E, "Advanced Engineering Mathematics", John Wiley and Sons, 10th Edition, New Delhi, 2016.