# MA3303 Probablity and Complex Functions Lecture Notes

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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.