__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.