This book provides an introduction into the mathematical concepts and tools necessary for understanding the theory of probability and the dynamics of stochastic processes central to a number of application areas in engineering sciences, biology medicine and finance. The material covered in the book, however, is particularly suited to an undergraduate programme in electronics and communications engineering, for it focuses on probability and the random variable, on random processes, linear systems and probabilistic tools for modelling of noise, which are of direct relevance to this branch of engineering. Each topic is introduced with the fundamental concepts and underlying theories in a concise manner, and is then followed up with several worked-out examples for developing problem-solving skills in the learner. Many of the problems have been drawn from previous years'' examination papers to give students an exposure to the variety and kinds typically encountered in exam situations; the focus of the selection is to train them in the use of explicit probability distributions for solving engineering and physics problems.

The contents are mapped to JNTU ECE syllabus, and many of the solved examples and exercises are drawn from previous JNTU question papers, making it a good match for JNTU market. However, since the coverage of topics is typical of any undergraduate programme on the subject, the book can be promoted everywhere in the country.

**Y Mallikarjuna Reddy**, Principal, Vasireddy Venkitadri Institute of Technology, Nambur, Guntur, is a professor in the department of Electronics and Communications Engineering. He has more than 22 years of teaching experience. He obtained his MTech degree from JNTU Kakinada in 1990 and his PhD in 2009 from Osmania University, Hyderabad. He is a member of I(E) and ISTE, India. He has published research papers in journals of international repute and is currently engaged in research in the areas of radar signal and image processing.

*Preface*

**1 Introduction to Probability**

1.1 Introduction

1.2 Set Theory

1.2.1 Terms of Set Theory

1.2.2 Set Operations

1.2.3 Laws of Sets

1.2.4 Sample Spaces

1.2.5 Events

1.3 The Relative Frequency and Axioms of Probability

1.3.1 Probability Introduced through Relative Frequency

1.3.2 Probability Introduced through Axioms

1.3.3 Classical Definition of Probability

1.4 Mathematical Model of Experiments

1.4.1 Examples of Experiments

1.5 Joint and Conditional Probability

1.5.1 Joint Probability

1.5.2 Conditional Probability

1.5.3 Properties of Conditional Probability

1.6 Total Probability Theorem

1.7 Bayes’ Theorem

1.8 Independent Events

1.8.1 Multiplication Theorem of Probability

1.8.2 Properties of Independent Events

1.9 Combined Sample Space

1.9.1 Independent Experiments

1.9.2 Permutations and Combinations

1.10 Bernoulli Trials

*Additional Problems*

*More Solved Examples*

*Questions*

*Problems*

*Answers*

*Multiple-Choice Questions*

*Answers*

**2 The Random Variable**

2.1 Introduction

2.1.1 Random Variable

2.1.2 Classifications of Random Variables

2.2 Probability Distribution Function

2.2.1 Expression for Distribution Function

2.3 Probability density function

2.3.1 Expression for Density Function

2.3.2 Properties of Probability Distribution Functions

2.3.3 Properties of Probability Density Functions

2.3.4 Probability Mass Function

2.4 Examples of Distribution and Density Functions

2.4.1 The Gaussian Density Function

2.4.2 Uniform Density Function

2.4.3 Exponential Probability Density Function

2.4.4 Rayleigh Probability Density Function

2.4.5 Binomial Probability Density Function

2.4.6 Poisson Probability Density Function

2.5 Conditional Distribution Function

2.5.1 Properties of Conditional Distribution Function

2.6 Conditional Density Function

2.6.1 Properties of Conditional Density Functions

2.7 Distribution Function for a Conditional Event

*Additional Problems*

*More Solved Examples*

*Questions *

*Problems *

*Answers *

*Multiple-Choice Questions*

*Answers *

**3 Operations on One Random Variable**

3.1 Introduction

3.2 Mathematical expectation

3.2.1 Expected Value of a Random Variable

3.2.2 Expected Value of a Function of a Random Variable

3.2.3 Conditional Expectation of a Random Variable

3.3 Properties of Expectation

3.4 Moments

3.4.1 Moments about the Origin

3.4.2 Moments about the Mean

3.5 Variance

3.5.1 Physical Significance of Variance and Standard Deviation

3.5.2 Skew and Coefficient of Skewness

3.5.3 Properties of Variance

3.5.4 Relationship between Central Moments and Moments about Origin

3.6 Functions for Moments

3.6.1 Characteristic Function

3.6.2 Properties of Characteristic Function

3.6.3 Moment Generating Function

3.6.4 Properties of Moment Generating Function

3.7 Inequalities

3.7.1 Chebychev’s Inequality

3.7.2 Markov Inequality

3.7.3 Chernoff’s Inequality & Bound

3.8 Transformations of a Random Variable

3.8.1 Monotonic Transformation of a Continuous Random Variable

3.8.2 Non-Monotonic Transformation of a Continuous Random Variable

3.8.3 Transformation of a Discrete Random Variable

*Additional Problems*

*More Solved Examples*

*Questions*

*Problems*

*Answers*

*Multiple-Choice Questions*

*Answers*

**4 Multiple Random Variables**

4.1 Introduction

4.2 Joint probability Distribution Function

4.2.1 Properties of Joint Distribution Function

4.3 Joint Probability Density Function

4.3.1 Properties of Joint Density Function

4.4 Conditional Distribution and Density Functions

4.4.1 Point Conditioning

4.4.2 Internal Conditioning

4.5 Statistical Independence of Random Variables

4.6 Sum of Random Variables

4.6.1 Two Random Variables

4.6.2 Multiple Random Variables 4.7 Central Limit Theorem

4.8 Probability Mass Function

*Additional Problems*

*More Solved Examples Questions *

*Problems *

*Answers *

*Multiple-Choice Questions 3*

*Answers*

**5 Operations on Multiple Random Variables** ** **

5.1 Introduction

5.2 Function of Joint Random Variables

5.3 Joint Moments

5.3.1 Joint Moments about the Origin

5.3.2 Correlation

5.3.3 Properties of Correlation

5.3.4 Joint Central Moments

5.3.5 Covariance

5.3.6 Correlation Coefficient

5.3.7 Properties of Covariance

5.4 Joint Characteristic Function

5.4.1 Properties of Joint Characteristic Function

5.5 Joint Moment Generating Function

5.5.1 Properties of Joint Moment Generating Function

5.6 Gaussian Random Variables

5.6.1 Two Random Variables

5.6.2 N Random Variables

5.6.3 Properties of Gaussian Random Variables

5.7 Transformation of Random Variables

5.8 Linear Transformation of Gaussian Random Variables

5.9 Conditional Gaussian Density Functions

*Additional Problems*

*More Solved Examples*

*Questions*

*Problems*

*Answers*

*Multiple-Choice Questions*

*Answers*

**6 Random Processes**

6.1 Introduction

6.2 Random Process

6.3 Classification of Random Processes

6.3.1 Continuous Random Processes

6.3.2 Discrete Random Processes

6.3.3 Continuous Random Sequencees

6.3.4 Discrete Random Sequencees

6.4 Distribution and Density Functions of Random Processes

6.4.1 Joint Distribution Functions of Random Process

6.4.2 Joint Density Functions of Random Process

6.5 Independent Random Processes

6.6 Statistical Properties of Random Processes

6.6.1 Mean

6.6.2 Autocorrelation

6.6.3 Cross Correlation

6.7 Stationary Processes

6.7.1 First-order Stationary Processes

6.7.2 Second-Order Stationary Processes

6.7.3 Wide-Sense Stationary Processes (WSS)

6.7.4 Jointly Wide-Sense Stationary Process

6.7.5 Strict-Sense Stationary Processes (SSS)

6.8 Time Averages of a Random Process

6.8.1 Time Average Function

6.8.2 Time Autocorrelation Function

6.8.3 Time Cross Correlation Function

6.9 Ergodic Theorem and Ergodic Processes

6.9.1 Ergodic Processes

6.9.2 Jointly Ergodic Processes

6.9.3 Mean Ergodic Processes

6.9.4 Autocorrelation Ergodic Processes

6.9.5 Cross Correlation Ergodic Processes

6.10 Properties of Autocorrelation Functions

6.11 Properties of Cross Correlation Functions

6.12 Covariance Functions for Random Processes

6.12.1 Autocovariance Function

6.12.2 Cross Covariance Function

6.13 Gaussian Random Processes

6.14 Poisson Random Processes

*Additional Problems*

*More Solved Examples*

*Questions *

*Problems *

*Answers *

*Multiple-Choice Questions*

*Answers *

**7 Random Processes: Spectral Characteristics**

7.1 Introduction

7.2 Power Density Spectrum

7.2.1 Average Power of the Random Process

7.3 Properties of the Power Density Spectrum

7.4 Bandwidth of the Power Density Spectrum

7.5 Cross Power Density Spectrum

7.5.1 Average Cross Power

7.6 Properties of Cross Power Density Spectrum

*Additional Problems*

*More Solved Examples*

*Questions*

*Problems*

*Answers*

*Multiple-Choice Questions*

*Answers*

**8 Linear Systems with Random Processes**

8.1 Introduction

8.2 Linear and Time Invariant Systems

8.2.1 Linear System

8.2.2 Response of the Linear System

8.2.3 Linear Time Invariant System

8.2.4 Transfer Function of LTI System

8.2.5 Causal Systems

8.2.6 Stable Systems

8.2.7 Ideal Systems

8.3 Response of Linear Systems to Random Signals

8.3.1 System Response

8.3.2 Mean Value of Output Response

8.3.3 Mean Square Value of Output Response

8.3.4 Autocorrelation Function of Output Response

8.3.5 Cross Correlation Function of Response

8.4 Spectral Characteristics of System Response

8.4.1 Power Density Spectrum of Response

8.4.2 Spectrum Bandwidth

8.5 Types of Random Processes

8.5.1 Lowpass Random Processes

8.5.2 Bandpass Random Processes

8.5.3 Band Limited Random Processes

8.5.4 Narrow Band Processes

8.5.5 Properties of Band-Limited Processes

8.6 Noise

8.6.1 Introduction

8.7 Classification of Noise

8.7.1 External Noise

8.7.2 Internal Noise

8.8 White Noise or White Gaussian Noise

8.8.1 Power Spectrum of White Noise

8.8.2 Band Limited White Noise

8.9 Resistor Noise Voltage

8.10 Equivalent Noise Resistor

8.11 Resistor Noise Spectral Density

8.12 Available Noise Power

8.13 Equivalent Noise Temperature

8.14 Noise through Two Port Networks

8.15 Signal-to-Noise Ratio

8.16 Available Power Gain

8.17 Equivalent Noise Bandwidth

8.18 Equivalent (Effective) Input Noise Temperature

8.19 Noise Figure

8.19.1 Noise Figure in Terms of Available Power Gain

8.19.2 Noise Figure in Terms of Input Noise Temperature

8.19.3 Noise Figure in Terms of Signal-to-Noise Ratio

8.19.4 Noise Figure in Terms of Network Transfer Function

8.19.5 Average Operating Noise Figure

8.20 Output Noise Power and System Noise Power

8.21 Noise in Cascade Amplifiers

8.22 Antenna Noise Temperature

8.23 Narrow Band Noise

8.23.1 In Phase and Quadrature Components of a Narrow Band Noise

8.24 Properties of a Narrow Band Noise

8.25 Ideal Narrow Band White Noise

*Additional Problems*

*More Solved Examples*

*Questions*

*Problems*

*Answers*

*Multiple-Choice Questions*

*Answers*

Solved JNTU Question Papers for the Year 1- 2

Appendix A: Indefinite Integrals, Definite Integrals and Finite Series

Appendix B: Fourier Transform Pairs

Bibliography