Entropy and information theory stanford ee stanford university. Formally, a random variable is a function that assigns a real number to each outcome in the probability space. This is an introduction to the main concepts of probability theory. An introduction to information theory dover books on. For a kary random variable, the entropy if maximized if px k1k, i. This site is the homepage of the textbook introduction to probability, statistics, and random processes by hossein pishronik. It teaches basic theoretical skills for the analysis of these objects, which include. Because of the importance of this subject, many universities added this syllabus in. The joint distribution of these two random variables is as follows. Today, we cover some of the basics of information theory. Used in studying chance events, it is defined so as to account for all possible outcomes of the event.
Then, by theorem \\pageindex1\ the expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables. The output from this channel is a random variable y over these same four symbols. Information theory is a subfield of mathematics concerned with transmitting data across a noisy channel. Random variable, in statistics, a function that can take on either a finite number of values, each with an associated probability, or an infinite number of values, whose probabilities are summarized by a density function. Im currently selfstudying and im looking for books focusing on random variables and their transformations, which possibly contain examples like the one in this question.
It selection from probability, random variables, and random processes. The text is concerned with probability theory and all of its mathematics, but now viewed in a wider context than that of the standard textbooks. The concept of information entropy was introduced by claude shannon in his 1948 paper a mathematical theory of. Download probability, random variables and stochastic processes by athanasios papoulis. Unnikrishna pillai the new edition of probability, random variables and stochastic processes has been updated significantly from the previous edition, and it now includes coauthor s. This is shannons entropy hx of the random variable x having distribution px. Probability theory and stochastic processes notes pdf ptsp pdf notes book starts with the topics definition of a random variable, conditions for a function to be a random variable, probability introduced through sets and relative frequency. We will show this in the special case that both random variables are standard normal.
It is well beyond the scope of this paper to engage in a comprehensive discussion of that. Lecture notes on information theory preface \there is a whole book of readymade, long and convincing, lavishly composed telegrams for all occasions. For further reading, the following book is recommended. It also includes applications in digital communications, information. Can anyone recommend good books on transformation of random variables and distributions. Statistics and probability overview of random variable. Chapter 3 is devoted to the theory of weak convergence, the related concepts of distribution and characteristic functions and two important special cases. Motivationinformation entropy compressing information information i let x be a random variable with distribution px. This book places particular emphasis on random vectors, random matrices, and random projections. This tract develops the purely mathematical side of the theory of probability, without reference to any applications. This book goes further, bringing in bayesian data modelling. Suppose x and y are two independent random variables, each with the standard normal density see example 5.
Mixed random variables, as the name suggests, can be thought of as mixture of discrete and continuous random variables. You see, what gets transmitted over the telegraph is not the text of the telegram, but simply the number under which it is listed in the book. More generally, this can be used to quantify the information in an event and a random variable, called entropy, and is calculated using probability. Who and when introduced the concept of random variable, was it a basic notion before measure theory.
Information theory, pattern recognition, and neural networks. Gray springer, 2008 a selfcontained treatment of the theory of probability, random processes. Probability, random variables, and random processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses. Checkout the probability and stochastic processes books for reference purpose. There will be a third class of random variables that are called mixed random variables. A cornerstone of information theory is the idea of quantifying how much information there is in a message. A random variable can take on many, many, many, many, many, many different values with different probabilities. How much do you really need to know and where do you start. In this article, we are providing the ptsp textbooks, books, syllabus, and reference books for free download. Information theory, inference, and learning algorithms. Lecture notes on probability theory and random processes. Entropy can be calculated for a random variable x with k in k discrete. Probability theory and stochastic processes notes pdf file download ptsp pdf notes ptsp notes. The emphasis throughout the book is on such basic concepts as sets, the probability measure associated with sets, sample space, random variables, information.
Probability, random variables, and random processes. The notion of entropy, which is fundamental to the whole topic of. A discrete time information source xcan then be mathematically modeled by a discretetime random process fxig. Pinskers classic information and information stability of random variables and processes and by the seminal work of a. Senetas paper is also interesting, but it rewrites everything, starting with bernoullis 17 ars conjectandi, in modern notation with random variables, so it is hard to tell where either they, or the notation, originated. It assumes little prior knowledge and discusses both information with respect to discrete and continuous random variables. Random processes in communication and control wikibooks. This book provides a systematic exposition of the theory in a setting which contains a balanced mixture of the classical approach and the modern day axiomatic approach. Probability, random processes, and ergodic properties.
The general case can be done in the same way, but the calculation is messier. Define your own discrete random variable for the uniform probability space on the right and sample to find the empirical distribution. Information theory studies the quantification, storage, and communication of information. The authors have comprehensively covered the fundamental principles, and have demonstrated. The same rules will apply to the online copy of the book as apply to normal books. Another way to show the general result is given in example 10. The set of all possible realizations is called support and is denoted by notation. A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete. Intuitively, the entropy hx of a discrete random variable x is a measure of the amount of uncertainty associated with the value of. Those in the disciplines of mathematics physics, and electrical engineering will find this book useful. Examples are entropy, mutual information, conditional entropy, conditional information, and relative entropy discrimination, kullbackleibler. A tutorial introduction is a highly readable first account of shannons mathematical theory of communication, now known as information theory.
Random process an event or experiment that has a random outcome. For any probability distribution, entropy is a quantity to capture the uncertainty of information of a random variable, which agrees with the intuitive notion of a measure of information. Sending such a telegram costs only twenty ve cents. It is intended for firstyear graduate students who have some familiarity with probability and random variables, though not. What are some good books for learning probability and statistics. Probability, random variables and stochastic processes was designed for students who are pursuing senior or graduate level courses, in probability. Functions of random variables, expectation, limit theorems 122 4. What is the best book for probability and random variables. A tutorial introduction, university of sheffield, england, 2014. A random variable is a numerical description of the outcome of a statistical experiment. Each lecture contains detailed proofs and derivations of all the main results, as well as solved exercises. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables.
Information theory this is a brief tutorial on information theory, as formulated by shannon shannon, 1948. A primer on shannons entropy and information bourbaphy. Probability theory and stochastic processes pdf notes. Beginning with a discussion on probability theory, the text analyses various types of random. We will discuss discrete random variables in this chapter and continuous random variables in chapter 4. The real number associated to a sample point is called a realization of the random variable. Probability theory and stochastic processes book link complete notes. Dobrushin on information measures for abstract alphabets and their convergence properties. Once you understand that concept, the notion of a random variable should become transparent see chapters 4 5. The entropy hx of a discrete random variable x with probability distribution. And it makes much more sense to talk about the probability of a random variable equaling a value, or the probability that it is less than or greater than something, or the probability that it has some property. Browse the amazon editors picks for the best books of 2019, featuring our. Random variables two methods to describe a random variable rv x. Which book is best for random variable and random process.
Probability laws and probability density functions of random vectors 64 6. This book presents the fundamental concepts of information theory in a. Now the book is published, these files will remain viewable on this website. The intent was and is to provide a reasonably selfcontained advanced treatment of measure theory, probability theory, and the theory of discrete time random processes with an. The information entropy, often just entropy, is a basic quantity in information theory associated to any random variable, which can be interpreted as the average level of information, surprise, or uncertainty inherent in the variable s possible outcomes. A rvp x is discrete if there exists a countable set x fx. Can anyone recommend good books on transformation of. Probability and random processes provides a clear presentation of foundational concepts with specific applications to signal processing and communications, clearly the two areas of most interest to students and instructors in this course it includes unique chapters on narrowband random processes and simulation techniques. I got it because of the entropy of continuous variables topic but read some more fantastic chapters like noisy channel. Information theory often concerns itself with measures of information of the distributions associated with random variables. Highdimensional probability is an area of probability theory that studies random objects in rn where the dimension ncan be very large. Probability theory and stochastic processes is one of the important subjects for engineering students. I we want to quantify the information provided by each possible outcome. Thousands of books explore various aspects of the theory.
Information theory, inference, and learning algorithms, 2003. Probability theory and stochastic processes books and. Pinskers classic information and information stability of random variables and processes and by the seminal. Probability random variables and stochastic processes. When originally published, it was one of the earliest works in the field built on the axiomatic foundations introduced by a. In rigorous measuretheoretic probability theory, the function is also required to be measurable see a more rigorous definition of random variable. The input source to a noisy communication channel is a random variable x over the four symbols a,b,c,d.
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