Contents

• 1. Basic Probability Theory

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  Objectives, Introduction, Events and Sample Space, Algebra of Events, Exhaustive Events, Mutually Exclusive Events, Equally Likely Events, Partition of the Sample Space S, Probability, The Axioms of Probability, Conditional Probability, Independent Events, Illustrative Examples, Exercise 1.1, Baye’s Theorem, Bernoulli Trials, Generalized Bernoulli Trials, Illustrative Examples, Exercise 1.2.
   
  2. Random Variables

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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, Illustrative Examples, Two Dimensional Random Variables, Probability Mass Function of (x, y), Cumulative Distribution Function, Marginal Probability Distribution, Conditional Probability Distribution, Independent Random Variables, Probability Density Function of (x, y), Joint Distribution Function, Marginal Density, Conditional Density, Independent Continuous Random Variables, Illustrative Examples, Exercise 2.2.
   
  3. Statistical Averages And Generating Functions

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  Objectives, Introduction, Expectation, Variance, Measures of Central Tendency and Dispersion, Skewness and Kurtosis, Measures of Central Tendency, Measures of Dispersion, Skewness, Kurtosis, Moments, 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, Probability Generating Function (pgf), Reliability, Failure Density and Hazard Function, Mean Time to Failure (MTTF), Illustrative Examples, Exercise 3.1, Expected Value of a Two Dimensional Random Variable, Moment of Bivariate Distribution, Conditional Expectation and Conditional Variance, Illustrative Examples, Exercise 3.2.
   
  4. Random Process

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  Objectives, Introduction, Classification and Description of A Random Process, Description of a Random Process, Definitions, Mean of a Random Process, Variance of the Stochastic Process, Autocorrelation of the Random Process, Autocovariance of the Random Process, Correlation Coefficient of the Random Process, Special Types of Random Processes, Markov Process, Independent Process, Process with Independent Increments, Renewal Process, Stationary Process, Bernoulli Process, Binomial Process, Poisson Process, Probability Law For Poisson Process, Properties of Poisson Process, Exercise 4.
   
  5. Theoretical Discrete Probability Distributions

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  Objectives, Introduction, Bernoulli Distribution, Mean and Variance of Bernoulli Distribution, Moment Generating Function, Probability Generating Function, Binomial Distribution, Mean and Variance of Binomial Distribution, Moments, Moment Generating Function and Recurrence Relation for Moments, Probability Generating Function, Mode of the Binomial Distribution, Fitting of Binomial Distribution (Recurrence Relation for the Probabilities of Binomial Distribution), Illustrative Examples, Exercise 5.1, Poisson Distribution, Mean and Variance of Poisson Distribution, Moments, Moment Generating Function and RecurrenceRelation for Moments, Probability Generating Function For Poisson Distribution, Mode of Poisson Distribution, Recurrence Relation for Probabilities of Poisson Distribution or Fitting of Poisson Distribution, Reproductive Property of Poisson Variate, Illustrative Examples, Exercise 5.2.
   
  6. Continuous Probability Distributions

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  Objectives, Introduction, 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, Fitting of Normal Distribution, Rectangular or Univorm Distribution, Moments and Moment Generating Function, Mean and Variance, Exponential Distribution, Moments, Moment Generating Function, Mean and Variance, Memoryless Property of Exponential Distribution, Illustrative Examples, Exercise 6.
   
  7. Correlation and Regression

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  Objectives, Introduction, Bivariate Distribution, Correlation, Measure of Correlation : Karl Pearson Coefficient of Correlation, Rank Correlation, Correlation of Bivariate Frequency Distribution, Curve Fitting, Principle of Least Squares, Fitting of a Straight Line, Fitting a Parabola, Fitting of Other Curves, Regression, Linear Regression, Lines of Regression, Angle Between Two Lines of Regression, Standard Error of Estimate or Residual Variance, Coefficient of Determination, Multiple Regression, Curvilinear Regression, Normal Correlation Analysis, Normal Regression Analysis, Illustrative Examples, Exercise 7.
   
  8. Queueing Theory

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  Objectives, Introduction, Elementary Queueing Process, Essential Features of a Queueing System, Performance Measures of a Queueing System, Transient and Steady State of the System, Notations, Pure Birth Process, Markovian Property of Inter arrival Time, Pure Death Process, Birth and Death Process, Kendall’s Notation, Queueing Model I M|M|1 : ?|FIFO, Queueing Cost Behaviour, Queueing Model II (M|M|1) : (N|FCFS), Illustrative Examples, Exercise 8.1, Queueing Model III, Queueing Model IV (M|M|s) : (?|FCFS), Queueing Model V (M|M|s : N|FCFS), Illustrative Examples, Exercise 8.2.
   
  9. Markov Analysis

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  Objectives, Introduction, Discrete Parameter Markov Chain, Transition Probability Matrix (tpm), Probability Distribution and n-Step Transition Probabilities, Steady State Distribution, ChapMan Kolmogorov Theorem, Classification of States of Markov Chain, The Recurrence Time Probability, M|G|1 Queueing Model (M | G | 1) : (? | GD), Discrete Parameter Birth-Death Process, Illustrative Examples, Exercise 9.
   
  10. Open Queueing Networks

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  Objectives, Introduction, Network of Queues, Open Queueing Networks, Jackson Network, Exercise 10.
   
  P. Papers