Contents
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1.1 Introduction to Random Variables 1.1 Introduction to Random Variables
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1.1.1 Motivation 1.1.1 Motivation
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1.1.2 Definition of a Random Variable and Probability 1.1.2 Definition of a Random Variable and Probability
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1.1.3 Function of a Random Variable, Expected Value 1.1.3 Function of a Random Variable, Expected Value
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1.1.4 Total Variation Distance 1.1.4 Total Variation Distance
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1.2 Multiple Random Variables 1.2 Multiple Random Variables
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1.2.1 Joint and Marginal Distributions 1.2.1 Joint and Marginal Distributions
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1.2.2 Independence and Conditional Distributions 1.2.2 Independence and Conditional Distributions
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This is precisely the desired conclusion. D This is precisely the desired conclusion. D
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1.2.3 Bayes' Rule 1.2.3 Bayes' Rule
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1.2.4 MAP and Maximum Likelihood Estimates 1.2.4 MAP and Maximum Likelihood Estimates
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1.3 Random Variables Assuming Infinitely many Values 1.3 Random Variables Assuming Infinitely many Values
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1.3.1 Some Preliminaries 1.3.1 Some Preliminaries
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1.3.2 Markov and Chebycheff Inequalities 1.3.2 Markov and Chebycheff Inequalities
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1.3.3 Hoeffding's Inequality 1.3.3 Hoeffding's Inequality
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1.3.4 Monte Carlo Simulation 1.3.4 Monte Carlo Simulation
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1.3.5 Introduction to Cramér's Theorem 1.3.5 Introduction to Cramér's Theorem
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One Introduction to Probability and Random Variables
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Published:August 2014
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Abstract
This chapter provides an introduction to probability and random variables. Probability theory is an attempt to formalize the notion of uncertainty in the outcome of an experiment. For instance, suppose an urn contains four balls, colored red, blue, white, and green respectively. Suppose we dip our hand in the urn and pull out one of the balls “at random.” What is the likelihood that the ball we pull out will be red? The chapter first defines a random variable and probability before discussing the function of a random variable and expected value. It then considers total variation distance, joint and marginal probability distributions, independence and conditional probability distributions, Bayes' rule, and maximum likelihood estimates. Finally, it describes random variables assuming infinitely many values, focusing on Markov and Chebycheff inequalities, Hoeffding's inequality, Monte Carlo simulation, and Cramér's theorem.
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