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Unformatted text preview: 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, Conditional Probability Density Function and Stirlings Formula. 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 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. However the user must acknowledge the present website www.mhhe.com/papoulis as the source of information. 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 i i PILLAI 4 Laplaces Classical Definition: The Probability of an event A is defined apriori 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 We can use above classical definition to determine the probability that a given number is divisible by a prime p . , outcomes possible of number Total to favorable outcomes of Number ) ( A A P = ....
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This note was uploaded on 10/16/2009 for the course EL el6303 taught by Professor Prof during the Spring '09 term at NYU Poly.
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