Lecture notes on information theory statistics, yale university. Probability distribution function pdf a mathematical description of a discrete random variable rv, given either in the form of an equation formula or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome. Finding the constant k given pdf of a random variable. A particularly important random variable is the canonical uniform random variable, which we will write. Then hx elog 21px log 2 e1px by applying jensen with the r. Hx the entropy of a random variable is not changed by repeating it and hence from 1. Discrete random variables discrete random variables can take on either a finite or at most a countably infinite set of discrete values for example, the integers. 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. On the entropy region of discrete and continuous random variables and network information theory conference paper pdf available in circuits, systems and computers, 1977. Binomial random variables, repeated trials and the socalled modern portfolio theory pdf 12.
Of course, there is a little bit more to the story. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class usually defined in terms of specified properties or measures, then. Formally, a random variable is a function that assigns a real number to each outcome in the probability space. A discrete time information source xcan then be mathematically modeled by a discrete time random process fxig.
For a discrete random variable the probability model lists the possible values the random variable takes and the probability which which it takes those values. Y, and z be a random variable taking values in z, called the noise variable. The value pxx is the probability that the random variable xtakes the value x. Information theory and coding the computer laboratory. Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables including discrete random variables that take values in. Be able to describe the probability mass function and cumulative distribution function using tables. For a discrete random variable x, itsprobability mass function f is speci ed by giving the. What this means in practice is that there is a certain probability for the random variable x to take on any particular value in the sample space. Statistical independence, discrete random variables. Information theory often concerns itself with measures of information of the distributions associated with random variables. Examples expectation and its properties the expected value rule linearity variance and its properties uniform and exponential random variables cumulative distribution functions normal random variables.
What is the pdf of a product of a continuous random variable and a discrete random variable. Cs 70 discrete mathematics and probability theory multiple. 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. Entropy and information theory stanford ee stanford university. Random variables in many situations, we are interested innumbersassociated with the outcomes of a random experiment.
Discrete random variable an overview sciencedirect topics. Despite this, these notes discuss order statistics, in particular the maximum and the minimum, of ndiscrete random variables. In particular, many of the theorems that hold for discrete random variables do not hold for continuous variables. The distribution function or cumulative distribution function or cdf of is a function such that. The example provided above is of discrete nature, as the values taken by the random variable are discrete either 0 or 1 and therefore the random variable is called discrete random variable. In probability theory, a probability density function pdf, or density of a continuous random variable, is a function whose value at any given sample or point in the sample space the set of possible values taken by the random variable can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. Shannon defined the entropy of a discrete time discrete alphabet random pro cess xn. The characteristics of a probability distribution function pdf for a discrete random variable are as follows. In case the probability density function exists, this can be written as.
Browse other questions tagged probability probability theory or ask your own question. Notes on order statistics of discrete random variables in stat 512432 we will almost always focus on the order statistics of continuous random variables. 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. Random variables, also those that are neither discrete nor continuous, are often characterized in terms of their distribution function. In rendering, discrete random variables are less common than continuous random variables, which take on values over ranges of continuous domains e. What is the joint entropy hx, y, and what would it be if the random variables x. Pdf on the entropy region of discrete and continuous.
In general though, the pmf is used in the context of discrete random variables random variables that take values on a countable set, while the pdf is used in the context of continuous random variables. What is the pdf of a product of a continuous random. In many cases the random variable is what you are measuring, but when it comes to discrete random variables, it is usually what you are counting. Definition of mathematical expectation functions of random variables some theorems. Note that these are theoretical distributions as opposed to empirical. One very common finite random variable is obtained from the binomial distribution. We will see that the expectation of a random variable is a useful property of the distribution that satis es an important property. Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. 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. According to shannons definition, given a discrete random variable x. Indeed, if we want to oversimplify things, we might say the following. Each probability is between zero and one, inclusive inclusive means to include zero and one. These two types of random variables are continuous random variables and discrete random variables. So for the example of how tall is a plant given a new fertilizer, the random variable is the height of the plant given a new fertilizer.
For any discrete random variable y, log 2 ey elog 2 y. The entropy hx of a discrete random variable x with probability distribution. For instance, a random variable describing the result of a single dice roll has the p. A discrete channel,z is a singleinput singleoutput system with input alphabet x and output alphabet y. For information theory, the fundamental value we are interested in for a random variable x is the entropy of x. Discrete random variables probability density function. Find materials for this course in the pages linked along the left.
Lecture 4 random variables and discrete distributions. This, like the standard deviation, is a way to quantify the amount that a random variable is spread out around its mean. Define your own discrete random variable for the uniform probability space on the right and sample to find the empirical distribution. Lecture notes probabilistic systems analysis and applied. Variables distribution functions for discrete random variables continuous random vari. We will then use the idea of a random variable to describe the discrete probability distribution, which is a. Even if information theory is considered a branch of communication the ory, it actually spans a. E logpx 1 the entropy measures the expected uncertainty in x. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. Probability distribution function pdf for a discrete. Know the bernoulli, binomial, and geometric distributions and examples of what they model. Maximum entropy probability distribution wikipedia. X time a customer spends waiting in line at the store infinite number of possible values for the random variable.
Joint pdf and joint cdf of a discrete and continuous random variables. For any input random variable x, the noise variable z is independent of x, and the output random variable y is given by y x,z. A primer on information theory, with applications to neuroscience. Capacity of a discrete channel as the maximum of its mutual information over. Random variables continuous random variables and discrete. Expectation, and distributions we discuss random variables and see how they can be used to model common situations. Part i is a rigorous treatment of information theory for discrete and continuous systems. In this lesson, the student will learn the concept of a random variable in statistics. We shall often use the shorthand pdf for the probability density func tion pxx. We also say that hx is approximately equal to how much information we learn on average from one instance of the random variable x. Recall that the variance of a sum of mutually independent random variables is the sum of the individual variances. Notes on order statistics of discrete random variables. The main object of this book will be the behavior of large sets of discrete random variables. We also introduce common discrete probability distributions.
Continuous random variables and probability density functions probability density functions. The probability density function pdf of a random variable is a function describing the probabilities of each particular event occurring. Chapter 3 discrete random variables and probability. Learnedmiller department of computer science university of massachusetts, amherst amherst, ma 01003 september 16, 20 abstract this document is an introduction to entropy and mutual information for discrete random variables. Cs 70 discrete mathematics and probability theory fall 2009 satish rao,david tse lecture 16 multiple random variables and applications to inference in many probability problems, we have to deal with multiple r. This video lecture discusses the concept of sample space, random variables. If we plot the cdf for our coinflipping experiment, it would look like the one shown in the figure on your right. Poisson random variable and probability density function. Discrete probability distributions let x be a discrete random variable, and suppose that the possible values that it can assume are given by x 1, x 2, x 3. Since this is posted in statistics discipline pdf and cdf have other meanings too.
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