Theory of probability measure theory, classical probability and stochastic analysis lecture notes by gordan zitkovic. The presentation of this chapter follows closely 6. R, we refer to measurable functions as random variables. Functions con be defined on a system of sets to take values in any given. The mathematical foundations of probability theory are exactly the same as those of lebesgue integration. Random variables are often designated by letters and. For example, if we think about intervals on the real line, the natural measure is the length of those intervals i. A random variable, or stochastic variable, is a quantity that is subject to random variation. For instance, we may define it as the sum of the sides pointing up, or else the multiplication of the sides.
In rigorous probability theory, the function is also required to be measurable a concept found in measure theory see a more rigorous definition of random vector. Anyway, i started reading the book stochastic differential equation by b. If it has as many points as there are natural numbers 1, 2, 3. Why is measure theory so important in probability theory. For instance, a random variable describing the result of a single dice roll has the p. The set of possible outcomes is called the sample space. What matters are the values a random variable can take and the associated cdf. A random variable can be defined based on a coin toss by defining numerical values for heads and tails.
In probability theory, there exist several different notions of convergence of random variables. The way i like to think of it is that it is a function that, in a sense, relieves the problem of dealing with nonnumerical elements by assigning each of them a real number or realvalued vector so that they can be compared on the real number line. It is a function giving the probability that the random variable x is less than or equal to x, for every value x. Is this a discrete random variable or a continuous random variable. Random variables and measurable functions as described in section 1. Discrete and continuous random variables video khan. Ho september 26, 20 this is a very brief introduction to measure theory and measure theoretic probability, designed to familiarize the student with the concepts used in a phdlevel mathematical statistics course.
Basic random variable question measure theory approach. So lets say that i have a random variable capital x. A classical example of a random event is a coin tossing. The most common example of a separable complete metric space is. The real vector associated to a sample point is called a realization of the random vector. Discrete random variables probability density function. A starter on measure theory and random variables in this chapter, we present in section i. 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. Stat 8501 lecture notes baby measure theory charles j. Probability density function pdf is a statistical expression that defines a probability distribution for a continuous random variable as. This distribution does not have a pdf, and it is totally mysterious from the masters level theory point of view. In measure theory we sometimes consider signed measures, whereby is no longer nonnegative, hence its range is 1. So the kind of integration that makes sense is a combination of integrating the density of continuous random variable plus adding a term corresponding to how a discrete random variable are treated. A random variable is a variable whose value is unknown, or a function that assigns values to each of an experiments outcomes.
Random variables can be defined in a more rigorous manner by using the terminology of measure theory, and in particular the concepts of sigmaalgebra, measurable set and probability space introduced at the end of the lecture on probability. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that. If we are sure or certain that the event will occur, we say that. As a result, we always end up having to complement the. Random variable definition of random variable by the free. One way to look at measure theory is that it is way of making such ad hoc combinations of integration and addition respectable. The technical axiomatic definition requires to be a sample space of a probability triple, see the measure theoretic definition the probability that takes on a value in a measurable set. Lets define random variable y as equal to the mass of a random animal selected at the new orleans zoo, where i grew up, the audubon zoo.
Contents part i probability 1 chapter 1 basic probability 3 random experiments sample spaces events the concept of probability the axioms. The description in this chapter is simplified in that we consider only discrete or continuous random variables. If a sample space has a finite number of points, as in example 1. A random variable is a set of uncertain outcomes, resulting from an event of a random process. And it is equal to well, this is one that we covered in the last video. A continuous random variable is one which takes an infinite number of possible values. Change of variables formula in measure theory hui december 16, 2012 let.
For a discrete random variable, the cumulative distribution function is found by summing up the probabilities. However, measure theory is much more general than that. Before data is collected, we regard observations as random variables x 1,x 2,x n this implies that until data is collected, any function statistic of the observations mean, sd, etc. Probability and random variables 11 probabilitytheory probability theory provides the mathematical rules for assigning probabilities to outcomes of random experiments, e. Then the probability density function pdf of x is a function fx such that for any two numbers a and b with a. There is no agreedupon measure of the size of a random variable. Measure theory together with x from an additive system on which is additive but not completely additive if x 2.
The rigorous definition of measure will be given later, but now we can recall the familiar from the elementary. The probability density function pdf of a random variable is a function describing the probabilities of each particular event occurring. On the other hand, books written for the engineering students tend to be fuzzy in their attempt to avoid subtle mathematical concepts. If x gives zero measure to every singleton set, and hence to every countable set, xis called a continuous random variable. Oksendal, and im having some problem in understanding. Thus, for instance, an ndimensional random vector x is a set of. The random variable as a function will determine how these outcomes are measured. For instance, if the random variable x is used to denote the outcome of a. For example, we may assign 0 to tails and 1 to heads. Physicists rely a lot on intuition, and there is sometimes a tendency to view all of this proof stuff as useless and unnecessary bookkeeping.
This simplification enables us to develop the theory of random variables almost without reference to measure theory. We also say that hx is approximately equal to how much information we learn on average from one instance of the random variable x. If what we pay attention as outcomes in the sample space are distances, we can measure the distances to a reference point, or in between the dice, in. Definition the formalization of this idea in modern probability theory kolmogorov 33, iii is to take a random variable to be a measurable function f f on a probability space x. The masters level recipe for nding the probability density function by di erentiating the df fails. Random variables may be discrete, continuous, or neither.
Probability density function pdf is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete. The set of probabilities likelihoods of all outcomes of the random variable is called a probability distribution. Probability distribution and entropy as a measure of. Random variable definition of random variable by the. In rigorous probability theory we get a much more clear, if poorly named, formulation of this concept. We say that the function is measurable if for each borel set b. The conditional entropy is a measure of how much uncertainty remains about the random variable x when we know the value of y. Every random variable can be written as a sum of a discrete random variable and a continuous random variable. Discrete and continuous random variables video khan academy. Continuous random variables are usually measurements. One of the main concepts from measure theory we need to be. A random measure is a locally finite transition kernel from a abstract probability space. The presentation of this material was in uenced by williams 1991.
Lecture notes on probability theory and random processes. For the definitions, let be a separable complete metric space and let be its borel algebra. Next, we define the concept of a random variable, and through it, the concept of. We will always use upper case roman letters to indicate a random variable to emphasize the fact that a random variable is a function and not a number. Department of mathematics, the university of texas at austin. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. Lebesgue measure being the measuretheoretic analog of ordinary length. This illustrates an important point in probability theory. Generally we try to construct the simplest probability space. Finally, rigorous probability with measure theory opens up the doors to many more. If the coin is fair then after ntrials, hoccurs approximately n2 times, and so does t.
The basic problem in measure theory is to prove the existence of a. Theory of probability university of texas at austin. Make the analogy to expectation of a discrete random variable. As an example, consider the demand for a specific model of car next month. Random variables are very confusing if you think about them too hard what does it mean for a variable to be random. Without measure theory means that you wont be able to introduce math\sigmamathalgebra, but otherwise you go more or less in the same way. Well, this random variable right over here can take on distinctive values. How to explain the difference between random variable and.
The technical axiomatic definition requires to be a sample space of a probability triple, see the measuretheoretic definition. Jan 19, 2015 measure theory for applied research class. Probability theory stanford statistics stanford university. Random measures can be defined as transition kernels or as random elements. In correspondence with general definition of a vector we shall call a vector random variable or a random vector any ordered set of scalar random variables. Ho september 26, 20 this is a very brief introduction to measure theory and measuretheoretic probability, designed to familiarize the student with the concepts used in a phdlevel mathematical statistics course. All random variables defined on a discrete probability. Y is the mass of a random animal selected at the new orleans zoo. Functions of a random variable generation of a random variable jointly distributed random variables scalar detection ee 278b. Let me try to answer this from the point of view of a theoretical physicist. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile.
The definition is as following according to the book of john b. A random variable is defined as a real or complexvalued function of some random event, and is fully characterized by its probability distribution. Im not new to the concept of random variable and i know the measure theory. E logpx 1 the entropy measures the expected uncertainty in x. In probability theory, a martingale is a sequence of random variables i. The uncertainty in a probability distribution of x can be measured by many. 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. Pugachev, in probability theory and mathematical statistics for engineers, 1984. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. So is this a discrete or a continuous random variable. Dec 18, 2017 without measure theory means that you wont be able to introduce math\sigmamathalgebra, but otherwise you go more or less in the same way.