Probability - Statistics and Random Processes

Title
Contents
Preface to the Third Edition
Preface to the First Edition
1. Probability Theory
	Random Experiment
	Mathematical or Apriori Definition of Probability
	Statistical or Aposteriori Definition of Probability
	Axiomatic Definition of Probability
	Conditional Probability
	Independent Events
	Worked Examples 1(A)
	Exercise 1(A)
	Theorem of Total Probability
	Bayes’ Theorem or Theorem of Probability of Causes
	Worked Examples 1(B)
	Exercise 1(B)
	Bernoulli’s Trials
	De Moivre–Laplace Approximation
	Generalisation of Bernoulli’s Theorem Multinomial Distribution
	Worked Examples 1(C)
	Exercise 1(C)
	Answers
2. Random Variables
	Discrete Random Variable
	Probability Function
	Continuous Random Variable
	Probability Density Function
	Cumulative Distribution Function (cdf)
	Properties of the cdf F(x)
	Special Distributions
	Discrete Distributions
	Continuous Distributions
	Worked Examples 2(A)
	Exercise 2(A)
	Two-Dimensional Random Variables
	Joint Probability Density Function
	Cumulative Distribution Function
	Properties of F(x, y)
	Marginal Probability Distribution
	Conditional Probability Distribution
	Independent RVs
	Random Vectors
	Worked Examples 2(B)
	Marginal Probability Distribution of X: {i, pi*}
	Marginal Probability Distribution of Y: { j, p*j}
	Exercise 2(B)
	Answers
3. Functions of Random Variables
	Functions of One Random Variable
	How to Find fy(y), when fx(x) is Known
	One Function of Two Random Variables
	Two Functions of Two Random Variables
	An alternative method to find the pdf of Z = g(X, Y )
	Workd Examples 3
	Exercise 3
	Answers
4. Statistical Averages
	Statistical Measures
	Measures of Central Tendency
	Mathematical Expectation and Moments
	Relation Between Central and Non-central Moments
	Dispersion
	Definisions
	The Coefficient of Variation
	Skewness
	Kurtosis
	Pearson’s Shape Coefficients
	Expected Values of a Two-Dimensional RV
	Properties of Expected Values
	Conditional Expected Values
	Properties
	Worked Examples 4(A)
	Exercise 4(A)
	Linear Correlation
	Correlation Coefficient
	Properties of Correlation Coefficient
	Rank Correlation Coefficient
	Worked Examples 4(B)
	Exercise 4(B)
	Regression
	Equation of the Regression Line of Y on X
	Standard Error of Estimate of Y
	Worked Examples 4(C)
	Exercise 4(C)
	Characteristic Function
	Properties of MGF
	Properties of Characteristic Function
	Cumulant Generating Function (CGF)
	Joint Characteristic Function
	Worked Examples 4(D)
	Exercise 4(D)
	Bounds on Probabilities
	Tchebycheff Inequality
	Bienayme’s Inequality
	Schwartz Inequality
	Cauchy-Schwartz Inequality
	Worked Examples 4(E)
	Exercise 4(E)
	Convergence Concepts and Central Limit Theorem
	Central Limit Theorem (Liapounoff’s Form)
	Central Limit Theorem (Lindeberg–Levy’s Form)
	Worked Examples 4(F)
	Exercise 4(F)
	Answers
5. Some Special Probability Distributions
	Introduction
	Special Discrete Distributions
	Mean and Variance of the Binomial Distribution
	Recurrence Formula for the Central Moments of the Binomial Distribution
	Poisson Distribution as Limiting Form of Binomial Distribution
	Mean and Variance of Poisson Distribution
	Mean and Variance of Geometric Distribution
	Mean and Variance of Hypergeometric Distribution
	Binomial Distribution as Limiting Form of Hypergeometric Distribution
	Worked Examples 5(A)
	Exercise 5(A)
	Special Continuous Distributions
	Moments of the Uniform Distribution U (a, b)
	Mean and Variance of the Exponential Distribution
	Memoryless Property of the Exponential Distribution
	Mean and Variance of Erlang Distribution
	Reproductive Property of Gamma Distribution
	Relation Between the Distribution Functions (cdf) of the Erlang Distribution with l = 1 (or Simple Gamma Distribution) and (Poisson Distribution)
	Density Function of the Weibull Distribution
	Mean and Variance of the Weibull Distribution
	Standard Normal Distribution
	Normal Probability Curve
	Properties of the Normal Distribution N(m, s)
	Importance of Normal Distribution
	Worked Examples 5(B)
	Exercise 5(B)
	Answers
6. Random Processes
	Classification of Random Processes
	Methods of Description of a Random Process
	Special Classes of Random Processes
	Average Values of Random Processes
	Stationarity
	Example of an SSS Process
	Analytical Representation of a Random Process
	Worked Examples 6(A)
	Exercise 6(A)
	Autocorrelation Function and its Properties
	Properties of R(t)
	Cross-Correlation Function and Its Properties
	Properties
	Ergodicity
	Mean-Ergodic Process
	Mean-Ergodic Theorem
	Correlation Ergodic Process
	Distribution Ergodic Process
	Worked Examples 6(B)
	Exercise 6(B)
	Power Spectral Density Function
	Properties of Power Spectral Density Function
	System in the Form of Convolution
	Unit Impulse Response of the System
	Properties
	Worked Examples 6(C)
	Exercise 6(C)
	Answers
7. Special Random Process
	Definition of a Gaussian Process
	Processes Depending on Stationary Gaussian Process
	Two Important Results
	Band Pass Process (Signal)
	Narrow-Band Gaussian Process
	Quadrature Representation of a WSS Process
	Noise in Communication Systems
	Thermal Noise
	Filters
	Worked Examples 7(A)
	Exercise 7(A)
	Poisson Process
	Probability Law for the Poisson Process {x(t)}
	Second-Order Probability Function of a Homogeneous Poisson Process
	Mean and Autocorrelation of the Poisson Process
	Properties of Poisson Process
	Worked Examples 7(B)
	Exercise 7(B)
	Markov Process
	Definition of a Markov Chain
	Chapman–Kolmogorov Theorem
	Classification of States of a Markov Chain
	Worked Examples 7(C)
	Exercise 7(C)
	Answers
8. Tests of Hypotheses
	Parameters and Statistics
	Sampling Distribution
	Estimation and Testing of Hypotheses
	Tests of Hypotheses and Tests of Significance
	Critical Region and Level of Significance
	Errors in Testing of Hypotheses
	One-Tailed and Two-Tailed Tests
	Critical Values or Significant Values
	Procedure for Testing of Hypothesis
	Interval Estimation of Population Parameters
	Tests of Significance for Large Samples
	Worked Examples 8(A)
	Exercise 8(A)
	Tests of Significance for Small Samples
	Student’s t-Distribution
	Properties of t-Distribution
	Uses of t-Distribution
	Critical Values of t and the t-Table
	Snedecor’s F-Distribution
	Properties of the F-Distribution
	Use of F-Distribution
	Worked Examples 8(B)
	Exercise 8(B)
	Chi-Square Distribution
	Properties of c2-Distribution
	Uses of c2-Distribution
	c2-Test of Goodness of Fit
	Conditions for the Validity of c2-Test
	c2-Test of Independence of Attributes
	Worked Examples 8(C)
	Exercise 8(C)
	Answers
9. Queueing Theory
	Symbolic Representation of a Queueing Model
	Difference Equations Related to Poisson Queue Systems
	Values of P0 and Pn for Poisson Queue Systems
	Characteristics of Infinite Capacity, Single Server Poisson Queue Model I [M/M/1): (∞/FIFO) model], when ln = l and mn = m (l < m)
	Relations Among E(Ns), E(Nq), E(Ws) and E(Wq)
	Characteristics of Infinite Capacity, Multiple Server Poisson Queue Model II [M/M/s): (∞/FIFO) model], When ln = l for all n(l < sm)
	Characteristics of Finite Capacity, Single Server Poisson Queue Model III [(M/M/1): (k/FIFO) Model]
	Characteristics of Finite Queue, Multiple Server Poisson Queue Model IV [(M/M/s): (k/FIFO) Model]
	Worked Examples 9
	Exercise 9
	Answers
10. Design of Experiments
	Aim of the Design of Experiments
	Some Basic Designs of Experiment
	Comparison of RBD and LSD
	Worked Examples 10
	Exercise 10
	Answers
Appendix: Important Formulae
Index