For a more extensive exposition, the reader is encouraged to check, for example, with 2. For our exposition, we will also depend on, of course, some measure theory, including the. Suppose the continuous random variables x and y have the following joint probability density function. Rohlins disintegration theorem article pdf available in discrete and continuous dynamical systems 327 july 2012 with 435 reads. Measure theory and integration are presented to undergraduates from the perspective of probability theory. Probability and conditional expectations bridges the gap between books on probability theory and statistics by providing the probabilistic concepts estimated and tested in analysis of variance, regression analysis, factor analysis, structural equation modeling, hierarchical linear models and analysis of qualitative data the authors emphasize the theory of conditional expectations that is. Measure, integral, and conditional expectation analysis group tu. Here we pull out some power tools from measure theory.
Unconditional expectations michele pi er, lse 11 february 2012 you are in t0 and you want to form an expectation of x, which is a random variable whose realization occurs in t2. Conditional expectation 4 of 17 where the last equality follows from the fact that x1a is gmeasurable. The emphasis of this book is on general properties of random processes rather than the speci c properties of special cases. The main di erence is that a sequence can converge to more than one limit. Martingales martingales stopped martingales the martingale convergence theorems applications. The definition and existence of conditional expectation. The conditional expectation or conditional mean, or conditional expected value of a random variable is the expected value of the random variable itself, computed with respect to its conditional probability distribution as in the case of the expected value, a completely rigorous definition of conditional expected value requires a complicated. We try another conditional expectation in the same example. The first chapter shows why measure theory is needed for the formulation of problems in probability, and explains why one would have been forced to invent lebesgue theory had it not already existed to contend with the paradoxes of large numbers. These lecture notes are intended for a firstyear graduatelevel course on measuretheoretic probability. In kolmogorovs formulation of the general form of this concept see below, the existence of a conditional expectation is an immediate consequence of the radon. Special optional slide for anyone who knows measure theory and doubts that c. However each two limits of the sequence have distance zero from each other, so this does not matter too much. Probability, random processes, and ergodic properties.
Let h be the family of all gmeasurable elements in l2. Information and conditional expectation the importance of conditional expectation updated assessment of future values taking expectation with respect to a certain measure di. Lets take a look at an example involving continuous random variables. One of the main concepts from measure theory we need to be. We then introduce the conditional expectation of random variables and discuss its properties. A categorical view of conditional expectation prakash panangaden school of computer science mcgill university applied category theory 8th april 2020 panangaden mcgill university overview and cones 8 april 2020 1 45. Therefore, x is a version of the conditional expectation exjg. Unlike a conserved quantity in dynamics, which remains constant in time, a martingales value can change. Conditional expectation conditional expectation properties of conditional expectation. Probability theory stanford statistics stanford university.
It is defined as an alternative probability measure conditioned on a particular value of a random variable. We close with the more common definition of conditional expectation found in most probability and measure theory texts, essentially property d above. Since information in probability theory is represented by. Introduction martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Ergodic theory lecture notes imperial college london. Given a pseudometric space p, there is an associated metric space m. Measure theory in this chapter, we will introduce the minimal requirements of measure theory which will be needed later. This thesis presents the conditional valueatrisk concept and combines an analysis that covers its application as a risk measure and as a vector norm. An alternative way to calculate the mean of the geomp. Conditional expectation measure theory harmonic measure infinite sequence index point these keywords were added by machine and not by the authors. Theorem 8 conditional expectation and conditional variance let x and y be random variables. Why would anyone in the world be interested in measure theory and probability who didnt have a background in pure math. Well apply a standard conditioning argument in the discrete case to compute en. Pdf measures and integrals in conditional set theory.
Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. For both areas of application the theory is revised in detail and examples are given to show how to apply the concept in practice. 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 statis. In probability theory, it is customary to use the symbol p for a probability measure. The conditional mean and variance have the following useful properties. This process is experimental and the keywords may be updated as the learning algorithm improves. Conditional probability and expectation 3 conditional expectation properties 4. The following properties of measures are easily checked. One key idea is the notion of conditional expectation. In particular, we introduce the concept of a random variable being measurable with respect to a given. Conditional expectation purdue math purdue university. Radonnikodym theorem and conditional expectation february, 2002 conditional expectation re. 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 statis. Notice that all we are doing with conditional expectations is the standard.
For example, the posts on expectation and variance are both written from a measure. Optimal prediction a probability distribution gives a measure of knowledge or believe about a random process of interest. Suppose we roll a pair of dice, but dont look immediately at the outcome. Lecture notes on measuretheoretic probability theory.
Introduction to conditional probabilities and expectations. Pdf conditional measures and conditional expectation. Recall the undergraduate definition of conditional probability associated with bayes rule. Measure, integral, and conditional expectation handout may 2, 2012. The theory of pseudometric spaces is much the same as the theory of metric spaces. The conditional expectation as an orthogonal projection 164.
Martingales are only brie y discussed in the treatment of conditional expectation. Let,f,p be a probability space and let g be a algebra contained in f. The notes are used during a course that had as one of its principal aims a swift introduction to measure theory as far as it is needed in modern probability, e. When new information decreases that ignorance, it changes our probabilities.