The **theory of probability,** applied extensively in all fields of engineering and physical sciences to model situations and outcomes, finds usage in fields as varied as social and behavioural sciences, biology, economics, management and business studies as well. This book, written to cater to an undergraduate engineering curriculum, explains the concepts and the mathematics of probability and stochastic processes to enable a student to solve practical problems with confidence. It covers probability axioms, conditional probability, special distributions, random variables, expectations, generating functions, operations on random variables, random processes and their temporal and structural characteristics and response of linear systems to random signals. Several solved examples illustrating the application of key concepts have been included in each chapter. This, together with the generous number of chapter-end exercises of varied levels of difficulty makes this book invaluable as a textbook on the subject.

**Pradip Kumar Ghosh**is professor of electronics and communications engineering at the Mody Institute of Technology and Science (Deemed University), Sikar, Rajasthan. He obtained his B Tech and M Tech degrees in radio physics and electronics from Calcutta University in 1989 and 1991 respectively, and his Ph.D. (Tech) from the same university in 1997 with a fellowship from the Council of Scientific and Industrial Research (CSIR), New Delhi. Dr Ghosh has about twelve years of teaching experience, having taught at the National Institute of Science and Technology (NIST), Berhampur, Orissa, St. Xavier''s College, Kolkata and Murshidabad College of Engineering and Technology, Berhampur, West Bengal, prior to his current assignment. He is the author of

*Principles of Electronic Communications: Analog and Digital*, published by Universities Press.

**1. Theory of Probability**

1.1 Introduction

1.2 Uncertainty and Relative Frequency

1.3 Elements of Set Theory

1.4 Probability and Sample Space

1.5 Conditional Probability and Joint Probability

1.6 Independence and Bayesâ€™ Theorem

1.7 Bernoulli Trials

Exercises

**2. Theory of Random Variables**

2.1 Random Variables: An Introduction

2.2 Probability Distribution

2.3 Conditional Distribution

Exercises

**3. Functional Transformation of One Random Variable**

3.1 Introduction

3.2 Functions of a Random Variable

3.3 Mean, Variance and Moments of Random Variables

3.4 Generating Function and Moments

3.5 Chebychevâ€™s Inequality

3.6 Characteristic Functions of Random Variable

Exercises

**4. Statistical Characteristics of Two or More Random Variables**

4.1 Introduction

4.2 One Function of Several Random Variables

4.3 Multidimensional Random Variables

4.4 Multidimensional Characteristic Function

4.5 Central Limit Theorem

4.6 Conditional Distribution and Density

Exercises

**5. Operations on Multivariate Random Variables**

5.1 Introduction

5.2 Expected Value of a Function of Random Variables

5.3 Joint Characteristic Function

5.4 Jointly Gaussian Random Variables

5.5 General Transformation of Multiple Random Variables

Exercises

**6. Correlation Theory of Random Process**

6.1 Introduction

6.2 Distribution and Density Function

6.3 Stationarity and Statistical Independence

6.4 Time Average and Ergodicity

6.5 Correlation Function and Its Properties

6.6 Measuring the Correlation Function of Random Processes

6.7 Discrete Correlation Function

6.8 Gaussian Random Process

6.9 Differentiation and Integration of Random Processes

Exercises

**7. Spectral Representation of Random Processes**

7.1 Introduction

7.2 Power Spectrum of Random Processes

7.3 Autocorrelation and Power Spectrum Relationship

7.4 Cross-Power Spectral Density

7.5 Relation between Cross-Power Spectral Density and Cross-Correlation

7.6 Noise Definitions

Exercises

**8. Response of Linear System to Random Signals**

8.1 Introduction

8.2 Mathematical Model of Physical Systems

8.3 System Response

8.4 Mean and Mean-Square Value of System Response

8.5 Autocorrelation Function of a Random Output Signal

8.6 Cross-Correlation Function between Input and Output Random Processes

8.7 Power Spectrum of Output Processes

8.8 Band-Limited and Narrowband Processes

8.9 Modelling of Noise Sources

8.10 Available Power Gain, Noise Figure and Equivalent Input Noise Temperature

Exercises

*Bibliography*

Index

Index