Cover

Title Page

Copyright Page

Contents

Preface

Acknowledgement

CHAPTER 1 RANDOM VARIABLES

Introduction

Random Experiment

Sample Space

Trial and Event

Probability

Axioms of Theory of Probability

Theorems

Conditional Probability

Baye’s Theorem

Random Variables

Discrete Random Variable

Continuous Random Variable

Probability Mass Function

Probability Density Function

Cumulative Distribution Function (CDF)

Mathematical Expectation or Mean of Random Variable

Variance of the Random Variable

Moments of the Random Variable

Moments Generating Function (MGF)

Properties

Standard Distributions

Discrete Distributions

Binomial Distribution

Assumptions for Binomial Distribution

Mean and Variance of a Binomial Distribution

Moment Generating Function (MGF) of a Binomial Distribution

Additive Property of Binomial Distribution

Poisson Distribution

Mean and Variance of Poisson Distribution

MGF of Poisson Distribution

Additive Property of Poisson Distribution

Geometric Distribution

Mean and Variance of Geometric Distribution

MGF Functions of Geometric Distribution

Memoryless (forgetfulness) Property of Geometric Distribution

Negative Binomial Distribution

MGF of Negative Binomial Distribution

Mean and Variance of NBD

Continuous Distributions

Uniform Distribution

Mean and Variance Uniform Distribution

MGF of Uniform Distribution

Exponential Distribution

Mean and Variance of Exponential Distribution

Moment Generating Function (MGF) of Exponential Distribution

Memoryless Property of Exponential Distribution [Forgetfulness Property]

Gamma Distribution (Erlang Distribution)

Mean and Variance of Gamma Distribution

Moment Generating Function (MGF) of Gamma Distribution

Additive Property

Weibull Distribution

Mean and Variance of Weibull Distribution

Moment Generating Function (MGF)

Exercise

Solved Two Mark Questions

CHAPTER 2 Two Dimensional Random Variables

Joint distributions—Marginal and conditional distributions

Two Dimensional Random Variable

Two Dimensional Discrete Random Variable

Two Dimensional Continuous Random Variable

Joint Probability Function

Joint Probability Density Function

Marginal Distribution Function

Marginal Density Function

Joint (Cumulative) Distribution Function

Conditional Distribution of a Random Variable

Independent Random Variables

Covariance

Properties

Correlation

Type of Correlation

Karl Pearson’s Co-efficient of Correlation

Rank Correlation

Repeated Rank Correlation

Regression

Regression Co-efficient

Transformation of Random Variables

Central Limit Theorem (CLT)

Applications of CLT

Exercise

Solved Two marks Questions

CHAPTER 3 Classification of Random Process

Classification of Random Process

Deterministic (or) Predictable Random Process

Non-Deterministic Random Process

Correlation

Mean of a Random Process

Auto Correlation of Random Process

Auto Covariance or Covariance Function of Random Process

Relation Between Mean, Auto Correlation and Auto Covariance

Cross Correlation

Cross Covariance

Relation Between Cross Correlation and Cross Covariance

Stationary Process

Strict Sense Stationary (SSS) Process

Jointly Stationary Process

Wide Sense Stationary (WSS) Process

Non-stationary (or) Evolutionary Process

Time Average and Ensemble Average

Markov Process

Classification of Markov Process

One Step Transition Probability

Homogeneous Markov Chain

Stochastic Matrix

Transition Probability Matrix (TPM)

Regular Matrix

N-Step Transition Probability

Chapman-Kolmogorov Theorem

Probability Distribution of the Process

Steady State (or) Stationary Distribution

Poisson Process

Probability function of Poisson process{X(t)}

Probability Law for Poisson Process

MGF of Poisson Process

Autocorrelation of the Poisson Process

Auto Covariance of Poisson Process

Correlation coefficient of poisson process

Properties of Poisson Process

Binomial Process

Normal(Gaussian) Process

Sine Wave Process

Random Telegraph Process

Mean and Autocorrelation of Telegraph Process

Exercise

Solved Two Mark Questions

CHAPTER 4 Queueing Theory

Introduction to Queueing Models

Characteristics of a Queueing System

Customer Behavior

Kendal’s Notation for Representing Queue Mode

Model 1: Single Server Poisson Queue Model (M/M/1) : (∞/FIFO)

Model 2: Multi Server Poisson Queue Model

Model 3: Finite Capacity, Single Server Queue(M/M/1) : (N/FCFS)

Model 4: Finite Capacity Multi Server Queue Model(M/M/C) : (N/FCFS).

Model 5: [Self Service Model]

Exercise

Solved Two Mark Questions

CHAPTER 5 Non Markovian Queues and Queue Networks

Birth and death Process

Probability Distribution of X(t)

Pure - Birth and Pure - Death Process

Probability Pn (t) of Pure - Birth Process

Probability Function of Pure - Death Process

Pollaczek - Khintchine (P-K) Formula

M/G/1 Queue

Various Formula for (M/ G / 1) : (∞ / GD)

Networks of Queues

Series Queues

Series Queues with Blocking

Jackson Network

Closed Jackson Networks

The Arrival Theorem

Mean Value Analysis

Exercise

Solved Two Marks Questions

Question Paper

Dr. R Pugalarsu is currently Associate Professor, Department of Mathematics, RMK Group of Institution, Chennai.