Discrete probability theory

Part 1 of book "Introduction to machine learning" provide with knowledge about: a taste of machine learning; probability theory; basic algorithms; density estimation; limit theorems; panzer windows; exponential families; maximum likelihood estimation;...
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Lecture Probability Theory  Lecture 3 provides knowledge of Random Variables. This chapter presents the following content: Distribution Function, continuoustype random variables, discretetype random variables,...
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Lecture Probability Theory  Lecture 5: Function of a Random Variable. Let Y be a random variable, discrete and continuous, and let g be a function from R to R, which we think of as a transformation. For example, Y could be a height of a randomly chosen person in a given population in inches, and g could be a function which transforms inches to centimeters, i.e. g(y) = 2.54 × y. Then W = g(Y) is also a random variable, but its distribution (pdf), mean, variance, etc. will differ from that of Y.
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In this chapter, we will focus on two random variables, but once you understand the theory for two random variables, the extension to n random variables is straightforward. We will first discuss joint distributions of discrete random variables and then extend the results to continuous random variables.
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Lecture Probability Theory  Lecture 11: Conditional Density Functions and Conditional Expected Values. In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values.
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Lecture Probability Theory  Lecture 18: Power Spectrum. In this chapter includes the following topics: Power Spectra and Linear Systems, Discrete – Time Processes, Matched Filter, Spectrum Estimation / Extension Problem.
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The paper deals with the robust state bounding estimation problem of stochastic control systems with discrete time Markovian jump. By using Lyapunov functional method and probability theory, we propose new sufficient conditions to guarantee robust state boundedness for the stochastic control systems. The conditions are derived in terms of linear matrix inequalities, which is simple and convenient for testing and application.
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In this lesson you will learn about discrete Probability. Then you will learn: Introduction to discrete probability, probability theory, Bayes’ theorem, expected value and variance。
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The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory and probability theory. In this course, we shall use it for both these purposes. In the binomial asset pricing model, we model stock prices in discrete time, assuming that at each step, the stock price will change to one of two possible values. Let us begin with an initial positive stock price S0. There are two positive numbers, d and u, with 0
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In this chapter, we study the mathematical structure of a simple oneperiod model of a financial market. We consider a finite number of assets. Their initial prices at time t = 0 are known, their future prices at time t = 1 are described as random variables on some probability space. Trading takes place at time t = 0. Already in this simple model, some basic principles of mathematical finance appear very clearly. In Section 1.2, we single out those models which satisfy a condition of market efficiency: There are no trading opportunities which yield a profit without any downside risk.
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text is designed for an introductory probability course at the university level for sophomores, juniors, and seniors in mathematics, physical and social sciences, engineering, and computer science. It presents a thorough treatment of ideas and techniques necessary for a firm understanding of the subject. The text is also recommended for use in discrete probability courses. The material is organized so that the discrete and continuous probability discussions are presented in a separate, but parallel, manner.
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This is the fifth book of examples from the Theory of Probability. This topic is not my favourite, however, thanks to my former colleague, Ole Jørsboe, I somehow managed to get an idea of what it is all about. The way I have treated the topic will often diverge from the more professional treatment. On the other hand, it will probably also be closer to the way of thinking which is more common among many readers, because I also had to start from scratch. The prerequisites for the topics can e.g. be found in the Ventus: Calculus 2 series, so I shall refer the reader to...
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This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory.
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