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1 TABLE OF CONTENTS PROBABILITY THEORY Lecture – 1 Basics Lecture – 2 Independence and Bernoulli Trials Lecture – 3 Random Variables Lecture – 4 Binomial Random Variable Applications and Conditional Probability Density Function Lecture – 5 Function of a Random Variable Lecture – 6 Mean, Variance, Moments and Characteristic Functions Lecture – 7 Two Random Variables Lecture – 8 One Function of Two Random Variables Lecture – 9 Two Functions of Two Random Variables Lecture – 10 Joint Moments and Joint Characteristic Functions Lecture – 11 Conditional Density Functions and Conditional Expected Values Lecture – 12 Principles of Parameter Estimation Lecture – 13 The Weak Law and the Strong Law of Large numbers

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2 STOCHASTIC PROCESSES Lecture – 14 Stochastic Processes - Introduction Lecture – 15 Poisson Processes Lecture – 16 Mean square Estimation Lecture – 17 Long Term Trends and Hurst Phenomena Lecture – 18 Power Spectrum Lecture – 19 Series Representation of Stochastic processes Lecture – 20 Extinction Probability for Queues and Martingales Note: These lecture notes are revised periodically with new materials and examples added from time to time. Lectures 1 11 are used at Polytechnic for a first level graduate course on “Probability theory and Random Variables”. Parts of lectures 14 19 are used at Polytechnic for a “Stochastic Processes” course. These notes are intended for unlimited worldwide use. Any feedback may be addressed to pillai@hora.poly.edu S. UNNIKRISHNA PILLAI
3 PROBABILITY THEORY 1. Basics Probability theory deals with the study of random phenomena, which under repeated experiments yield different outcomes that have certain underlying patterns about them. The notion of an experiment assumes a set of repeatable conditions that allow any number of identical repetitions. When an experiment is performed under these conditions, certain elementary events occur in different but completely uncertain ways. We can assign nonnegative number as the probability of the event in various ways: ), ( i P ξ i ξ i ξ PILLAI

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4 Laplace’s Classical Definition: The Probability of an event A is defined a-priori without actual experimentation as provided all these outcomes are equally likely . Consider a box with n white and m red balls. In this case, there are two elementary outcomes: white ball or red ball. Probability of “selecting a white ball” , outcomes possible of number Total to favorable outcomes of Number ) ( A A P = . m n n + = (1-1) PILLAI
5 Relative Frequency Definition : The probability of an event A is defined as where n A is the number of occurrences of A and n is the total number of trials. The axiomatic approach to probability, due to Kolmogorov, developed through a set of axioms (below) is generally recognized as superior to the above definitions, as it provides a solid foundation for complicated applications.

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