The third edition features material on descriptive statistics. The value pxx is the probability that the random variable xtakes the value x. A discrete random variable is finite if its list of possible values has a fixed finite number of elements in it for example, the number of smoking ban supporters in a random sample of 100 voters has to be between 0 and 100. Discrete and continuous random variables video khan. Discrete random variable an overview sciencedirect topics.
It could be 1992, or it could be 1985, or it could be 2001. For instance, a random variable describing the result of a single dice roll has the p. If the random variable b is the outcome of a bernoulli experiment, and the probability of a successful outcome of b is p, we say b comes from a bernoulli distribution with success probability p where. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. A probability distribution is a table of values showing the probabilities of various outcomes of an experiment for example, if a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3. Because high entropy of a random sequence is a necessary condition of its use in cryptography, several general methods that increase the sequence entropy have. For example, if a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3.
There are discrete values that this random variable can actually take on. The concept of information entropy was introduced by claude shannon in his 1948 paper a mathematical theory of communication. More generally, this can be used to quantify the information in an event and a random variable, called entropy, and is calculated using probability. Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function cdf increases only by jump discontinuitiesthat is, its cdf increases only where it jumps to a higher value, and is constant between those jumps. The concept of random variable is central to the probability theory and also to rendering more specifically. 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. Data can be understood as the quantitative information about a. A discrete random variable is often said to have a discrete probability distribution. The output from this channel is a random variable y over these same four symbols. This section covers discrete random variables, probability distribution, cumulative distribution function and probability density function. Discrete random variables probability density function.
Entropy is a measure of the uncertainty in a random variable. In statistics, numerical random variables represent counts and measurements. In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. Gray springer, 2008 a selfcontained treatment of the theory of probability, random processes.
Examples are entropy, mutual information, conditional entropy, conditional information, and relative entropy discrimination, kullbackleibler. Random variables contrast with regular variables, which have a fixed though often unknown value. Random variables usually it is more convenient to associate numerical values with the outcomes of an experiment than to work directly with a nonnumerical description such as red ball. Information theory is a subfield of mathematics concerned with transmitting data across a noisy channel. Expected value of discrete random variables statistics. For information theory, the fundamental value we are interested in for a random variable x is the entropy of x. Introduction to probability theory and statistical inference. In the field of information theory, a quantity cal. Its value is a priori unknown, but it becomes known once the outcome of the experiment is realized. Nov 15, 2012 an introduction to discrete random variables and discrete probability distributions.
The entropy, h, of a discrete random variable x is a measure of the amount of uncertainty associated with the value of x. Introduction to discrete random variables and discrete. A probability distribution is a table of values showing the probabilities of various outcomes of an experiment. A random variable is a variable whose value depends on the outcome of a probabilistic experiment. Upper case letters such as x or y denote a random variable. A spinner in the shape of a regular hexagon is shown on the right. More generally, this can be used to quantify the information in an event and a random variable, called entropy, and is calculated. High school mathematics extensionsdiscrete probability.
Note however that the points where the cdf jumps may form a dense. For a discrete probability distribution of asset values, where losses are certain, the epd is the expectation of assets being less than losses. Mutual information between discrete variables with many. The method uses an auxiliary table and a novel theorem that concerns the entropy of a sequence in which the elements are a bitwise exclusiveor sum of independent discrete random variables. Difference between discrete and continuous variable with. Notes on order statistics of discrete random variables. Discrete random variables mathematics alevel revision. A random variable x x, and its distribution, can be discrete or continuous. Expected value of a random variable chapter 3 basic concepts of information theory. 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 variables possible outcomes. A random variable is a variable that takes on one of multiple different values, each occurring with some probability. A variable that assumes only values in a discrete set, such as the integers. A particularly important random variable is the canonical uniform random variable, which we will write. A random variable is a variable taking on numerical values determined by the outcome of a random phenomenon.
This book cover basic probability theory, random variables, random process, theoretical continuous discrete probability distributions, correlation and regression, queueing theory. Information theory is based on probability theory and statistics. To find the expected value of \y\, it is helpful to consider the basic random variable associated with this experiment, namely the random variable \x\ which represents the random permutation. When spun it eventually lands with one edge flat against the surface it is on.
The joint distribution of these two random variables is as follows. Introduction to discrete random variables introduction. In particular, as we discussed in chapter 1, sets such as n, z, q and their subsets are countable, while sets such as nonempty intervals a, b in r are uncountable. Cramerrao bounds for variance of estimators, twosample inference procedures, bivariate normal probability law, fdistribution, and the analysis of variance and nonparametric procedures. An informationtheoretical measure of the degree of indeterminacy of a random variable.
A random variable that may assume only a finite number or an infinite sequence of values is said to be discrete. Discrete random variable definition of discrete random. So, for example, the probability that will be equal to is and the probability that will be. Recall that discrete data are data that you can count. A mutual information estimator for continuous and discrete. Discrete probability functions and distribution 217. In the field of information theory, a quantity called entropy is used as a measure of information. Discusses probability theory and to many methods used in problems of statistical inference. Used in studying chance events, it is defined so as to account for all possible outcomes of the event. A tutorial introduction is a highly readable first account of shannons mathematical theory of communication, now known as information theory. 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. Entropy free fulltext a novel method for increasing.
It assumes little prior knowledge and discusses both information with respect to discrete and continuous random variables. A few examples of discrete and continuous random variables are discussed. Variable refers to the quantity that changes its value, which can be measured. Discrete random variables definition brilliant math. It wont be able to take on any value between, say, 2000 and 2001.
The first chapter of this lesson will be dedicated to introducing, explaining and understanding what a random variable is. Let be a random variable that can take only three values, and, each with probability. One very common finite random variable is obtained from the binomial distribution. This work is produced by the connexions project and licensed under the creative commons attribution license y abstract this module introduces the probability distribution unctionf pdf and its characteristics. Information theory often concerns itself with measures of information of the distributions associated with random variables. If the possible outcomes of a random variable can be listed out using a finite or countably infinite set of single numbers for example, 0.
The problem of maximizing the entropy of a sequence of independent, discrete random variables is considered mainly by scientists involved in the theory and practice of random numbers. Indeed, if we want to oversimplify things, we might say the following. There are several types of random variables, and the articles in the statistics section, on discrete and continuous probability distributions, provide detailed descriptions of them. Let x be a discrete random variable that takes values in the set x often referred to as the alphabet and has probability mass function px x px x, the entropy hx of the discrete random variable x is defined by if the logarithm has base 2, then hx has units bits. Discrete random variable synonyms, discrete random variable pronunciation, discrete random variable translation, english dictionary definition of discrete random variable. The main object of this book will be the behavior of large sets of discrete random variables. It is intended for firstyear graduate students who have some familiarity with probability and random variables, though not necessarily of random. The input source to a noisy communication channel is a random variable x over the four symbols a,b,c,d.
The theory of probability is a major tool that can be used to explain and understand the various phenomena in different natural, physical and social sciences. Let nx i,y j denote the number of samples with x i and y j values, and n t be the total number of samples. An information theoretical measure of the degree of indeterminacy of a random variable. A cornerstone of information theory is the idea of quantifying how much information there is in a message. An introduction to information theory by fazlollah m.
If you lose, add the amount that you last bet to the end of your list. 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. 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. Its support is and its probability mass function is. Introduction to discrete random variables introduction to. Discrete and continuous random variables video khan academy. In rendering, discrete random variables are less common than continuous random variables, which take on values over ranges of continuous domains e. The probability density function pdf of a random variable is a function describing the probabilities of each particular event occurring. Entropy free fulltext a novel method for increasing the. The values of a random variable can vary with each repetition of an experiment.
In this case the probability p i associated to z i can be written on the basis of the probability density function of z as p i fz i cover and thomas 1 show that the discrete entropy of the quantized variable z. Apr 19, 2019 if the random variable b is the outcome of a bernoulli experiment, and the probability of a successful outcome of b is p, we say b comes from a bernoulli distribution with success probability p where. Example what is the probability mass function of the random variable that counts the number of heads on 3 tosses of a fair coin. Next youll find out what is meant by a discrete random variable. A key idea in probability theory is that of a random variable, which is a variable whose value is a numerical outcome of a random phenomenon, and its distribution. If is a discrete random variable defined on a probability space and assuming values with probability distribution, then the entropy is defined by the formula. Probability theory and stochastic processes pdf notes sw. A random variable is discrete if its range is a countable set. Probability theory and stochastic processes pdf notes. A random variable is a function from a probability space to the real numbers. Probability distribution function pdf for a discrete random variable susan dean barbara illowsky, ph. Discrete probability distribution definition of discrete. An introduction to discrete random variables and discrete probability distributions.
In this paper, we propose a novel method for increasing the entropy of a sequence of independent, discrete random variables with arbitrary distributions. 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. For quantitative representation of average information per symbol we make the following assumptions. For instance, if the random variable x is used to denote the. The former refers to the one that has a certain number of values, while the latter implies the one that can take any value between a given range. Books statistics and probability theory books buy online.
Is this a discrete or a continuous random variable. Solvency measurement for propertyliability riskbased capital applications. When there are a finite or countable number of such values, the random variable is discrete. Best book of statistics and probability theory book buy online. A little like the spinner, a discrete random variable is a variable which can take a number of possible values. There are six possible outcomes of \x\, and we assign to each of them the probability \16\ see table \\pageindex3\. For a discrete random variable x, itsprobability mass function f is speci ed by giving the values fx px x for all x in the range of x. The probability distribution of a random variable x x tells us what the possible values of x x are and what probabilities are assigned to those values. The cumulative distribution function fy of any discrete random variable y is the probability that the random variable takes a value less than or equal to y. 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. Probability, random variables, and random processes. Discrete random variables probability density function pdf.
A random variable describes the outcomes of a statistical experiment in words. Statistics and probability overview of random variable. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information. Of course, there is a little bit more to the story. Well, that year, you literally can define it as a specific discrete year. Consider two discrete variable x and y with x 1, x 2, x n, and y 1, y 2, y m distinct values or categories, respectively.
912 726 1371 1531 895 1247 751 1609 146 1364 293 1009 525 945 127 1074 632 73 1249 1322 1561 1217 789 935 1267 654 907 1488 26 844