Contents

• 1. Probability

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  Objectives, Introduction, Sets, Equal Sets, Subsets and its Properties, Proper Subset, Universal Set, Null Set and Singlet, Finite Set and its Cardinality, Countable and Uncountable Sets, Complement of a Set, Difference of Two Sets, Power Set, Disjoint and Comparable Sets, Venn Diagrams, Algebra of Sets, Union Operation, Intersection Operation, Distributive Properties, Duality, Principle of Duality, Partition, Event and Sample Space, Exhaustive Events, Mutually Exclusive Events, Equally Likely Events, Probability, The Axioms of Probability, Basic Counting Principles, Permutation and Combination, Basic Counting Principles, Permutation, Combination, Conditional Probability, Joint Probability, Independent Events, Baye’s Theorem, Application of Bayes Theorem in Communication Systems, Bernoulli Trials, Generalized Bernoulli Trials, Illustrative Examples, Exercise.
   
  2. Random Variables and Statistical Averages

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  Objectives, Introduction, Random Variable, Discrete Random Variable, Continuous Random Variable, Probability Distribution of A Discrete Random Variable, Probability Mass Function, Distribution Function, Probability Distribution of a Continuous Random Variable, Probability Density Function, Distribution Function, Conditional Distribution and Density Function, Illustrative Examples, Exercise 2.1, Expectation, Variance, Measures of Central Tendency and Dispersion, Skewness and Kurtosis, Measures of Central Tendency, Measures of Dispersion, Skewness, Kurtosis, Moments and Moment Generating Function (mgf), Moment About Origin, Moment About Mean or Central Moment, Moment About An Arbitrary Point, Karl Pearson ? and ? Coefficents, Moment Generating Function (mgf), Cumulant Generating Function, Characteristic Function, Probability Generating Function (pgf), Reliability, Failure Density and Hazard Rate Function, Mean Time to Failure (MTTF), Inter Connection of Systems, Illustrative Examples, Exercise 2.2.
   
  3. Special Probability Distributions

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  Objectives, Introduction, Bernoulli Distribution, Mean and Variance of Bernoulli Distribution, Moment Generating Function, Probability Generating Function, Characteristic Function, Binomial Distribution, Mean and Variance of Binomial Distribution, Moments, Moment Generating Function and Recurrence Relation for Moments, Probability Generating Function, Characteristic Function, Mode of the Binomial Distribution, Fitting of Binomial Distribution (Recurrence Relation for the Probabilities of Binomial Distribution), Poisson Distribution, Mean and Variance of Poisson Distribution, Moments, Moment Generating Function and Recurrence Relation for Moments, Probability Generating Function For Poisson Distribution, Characteristic Function, Mode of Poisson Distribution, Recurrence Relation for Probabilities of Poisson Distribution or Fitting of Poisson Distribution, Reproductive Property of Poisson Variate, Illustrative Examples, Exercise 3.1, Rectangular or Uniform Distribution, Moments and Moment Generating Function, Mean and Variance, Characteristic Function, Normal Distribution, Standard Form of the Normal Distribution, Normal Probability Integral (Area Under the Standard Probability Curve), Mean and Variance of Normal Distribution, Moment Generating Function of Normal Distribution, Recurrence Relation for Even Order Central Moments, Characteristic Function, Fitting of Normal Distribution, Exponential Distribution, Moments, Moment Generating Function, Mean and Variance, Memoryless Property of Exponential Distribution, Rayleigh Distribution, One Function of One Random Variable, Illustrative Examples, Exercise 3.2.
   
  4. Multiple Random Variables

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  Objectives, Introduction, Vector Random Variable, Joint Probability Function, Joint Probability Mass Function, Joint Probability Density Function, Joint Distribution Function, For Discrete Random Variable, For Continuous Random Variable, Marginal Probability Function and Marginal Distribution Function, Marginal pmf and cdf, Marginal pdf and cdf, Conditional Probability Function and Distribution Function, For Discrete Random Variable, Conditional Density and Distribution Function, Statistical Independence, Expected Values of Multiple Random Variables, Variance or Covariance for Joint Distributions, Conditional Expectation and Conditional Variance, Moments of Bivariate Probability Distribution, Joint Moment Generating Function, Joint Characteristic Function, Distribution and Density Function of Sum of Random Variables, Functions of Random Variables, One Function of two Random Variables, Two Functions of Two Random Variables, Linear Transformation, Illustrative Examples, Exercise 4.1, Markov Inequality, Chebyshev’s Inequality, Cauchy Schwartz Inequality, Generalised Form-Bienayme Chebyschev’s Inequality, Central Limit Theorem, Exercise 4.2.
   
  5. Stochastic Processes and Linear Systems

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  Objectives, Introduction, Stochastic Process, Classification of a Random Processes, Description of a Random Process, Analytical Description, Joint Distribution Function, Statistics of a Random Process (Mean, Autocorrelation), Mean of a Random Process, Autocorrelation of the Random Process, Autocovariance of the Random Process, Correlation Coefficient of the Random Process, Stationary Process, First Order Stationary Process, Second Order Stationary Process, n-order and Strict Sense Stationary, Strict Sense Stationary Process (SSS), Wide sense Stationary Process, Properties of Autocorrelation Function, Properties of Cross Correlation Function, Statistical Independence, Time Averages, Ergodicity, Mean Ergodic Process, Correlation Ergodic Process, Random Processes and Linear Systems, Linear System, Characteristics of System Response, Illustrative Examples, Exercise.
   
  6. Special Random Processes and Linear Systems

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  Objectives, Introduction, Power Spectral Density (PSD), Properties of Power Spectral Density Function, Cross Power Spectral Density, Power Spectral Density of System Response, Band Pass Process (Signal), White Noise, Gaussian Random Process, Properties of the Gaussian Random Process, Process Depending on Stationary Gaussian Process, Narrow Band Gaussian Process, Illustrative Examples, Exercise.
   
  A. Appendix
  P. Paper