Anna University, Chennai

Department of Electrical and Electronics Engineering

Subject Code: MA3303

Subject Name: Probability and Complex Functions

Lecture Notes - All Units

Unit 1: Probability Theory

1.1 Introduction to Probability

Definition of Probability

Sample Space, Events, and Probability Space

Laws of Probability: Addition Law, Multiplication Law, Conditional Probability

1.2 Random Variables and Probability Distributions

Definition of Random Variables

Discrete and Continuous Random Variables

Probability Mass Function (PMF) and Probability Density Function (PDF)

Cumulative Distribution Function (CDF)

1.3 Expectation and Variance

Mean and Expected Value of a Random Variable

Variance and Standard Deviation

Properties of Expectation and Variance

Unit 2: Complex Numbers

2.1 Introduction to Complex Numbers

Definition and Representation of Complex Numbers

Algebraic Operations: Addition, Subtraction, Multiplication, Division

Polar Form and Exponential Form of Complex Numbers

2.2 Complex Functions

Definition of Complex Functions

Analytic Functions and Cauchy-Riemann Equations

Elementary Complex Functions: Exponential, Trigonometric, Logarithmic Functions

2.3 Complex Integration

Line Integrals and Contour Integrals

Cauchy's Integral Theorem and Integral Formula

Applications of Complex Integration

Unit 3: Probability Distributions

3.1 Discrete Probability Distributions

Bernoulli Distribution

Binomial Distribution

Poisson Distribution

3.2 Continuous Probability Distributions

Uniform Distribution

Normal (Gaussian) Distribution

Exponential Distribution

3.3 Joint Probability Distributions

Joint PMF and Joint PDF

Marginal and Conditional Distributions

Covariance and Correlation Coefficient

Unit 4: Complex Analysis

4.1 Complex Differentiation

Definition of Complex Differentiation

Cauchy-Riemann Equations and Analytic Functions

Complex Power Series and Taylor Series Expansion

4.2 Complex Integration Techniques

Contour Integration and Cauchy's Integral Theorem

Residue Theorem and Cauchy's Integral Formula

Applications of Complex Integration in Engineering Problems

Unit 5: Special Functions

5.1 Gamma Function

Definition and Properties of Gamma Function

Relation with Factorials and Binomial Coefficients

Applications in Probability and Statistical Distributions

5.2 Zeta Function

Definition and Properties of Zeta Function

Analytic Continuation and Functional Equation

Applications in Number Theory and Complex Analysis

These lecture notes provide comprehensive coverage of Probability Theory and Complex Functions, tailored for students in the Department of Electrical and Electronics Engineering at Anna University, Chennai. With a total of 61 pages in PDF format, these notes include detailed explanations, mathematical derivations, practical examples, and applications to facilitate understanding and application. Students will gain proficiency in probability concepts, complex number theory, and their applications in engineering and scientific problems.

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