Experiment (probability theory)
In probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: 1. * A sample space, , which is the set of all possible outcomes. 2. * An event space, which is a set of events , an event being a set of outcomes in the sample space. 3. * A probability function, which assigns each event in the event space a probability, which is a number between 0 and 1. In order to provide a sensible model of probability, these elements must satisfy a number of axioms, detailed in this article. In the example of the throw of a standard die, we would take the sample space to be . For the event space, we could simply use the set of all subsets of the sample space, which would then contain simple events such as ("the die lands on 5"), as well as complex events such as ("the die lands on an even number"). Finally, for the probability function, we would map each event to the number of outcomes in that event divided by 6 — so for example, would be mapped to , and would be mapped to . When an experiment is conducted, we imagine that "nature" "selects" a single outcome, , from the sample space . All the events in the event space that contain the selected outcome are said to "have occurred". This "selection" happens in such a way that if the experiment were repeated many times, the number of occurrences of each event, as a fraction of the total number of experiments, would most likely tend towards the probability assigned to that event by the probability function . The Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. In modern probability theory there are a number of alternative approaches for axiomatization — for example, algebra of random variables. (Wikipedia).
Probabiilty spaces, events and conditional probabilities | Probability and Statistics
We now introduce some more formal structures to talk about probabillities: first the idea of a sample space S--the possible outcomes of an experiment, and then the idea of a probability measure P on such a sample space. Together these two (S,P) make what we call a probability space. An e
From playlist Probability and Statistics: an introduction
(PP 6.3) Gaussian coordinates does not imply (multivariate) Gaussian
An example illustrating the fact that a vector of Gaussian random variables is not necessarily (multivariate) Gaussian.
From playlist Probability Theory
Random variables, means, variance and standard deviations | Probability and Statistics
We introduce the idea of a random variable X: a function on a probability space. Associated to such a function is something called a probability distribution, which assigns probabilities, say p_1,p_2,...,p_n to the various possible values of X, say x_1,x_2,...,x_n. The probabilities p_i h
From playlist Probability and Statistics: an introduction
(PP 5.1) Multiple discrete random variables
(0:00) Definition of a random vector. (1:50) Definition of a discrete random vector. (2:28) Definition of the joint PMF of a discrete random vector. (7:00) Functions of random vectors. A playlist of the Probability Primer series is available here: http://www.youtube.com/view_play_list?p=
From playlist Probability Theory
Probability & Statistics (8 of 62) The Probability Function - A First Look
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is the probability function. http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 Next video in series: http://youtu.be/zReGHNdWvIo
From playlist Michel van Biezen: PROBABILITY & STATISTICS 1 BASICS
Probability notation and terms, When you have equally likely outcomes, Conditional probability
Probability notation and terms, When you have equally likely outcomes, Conditional probability
From playlist Exam 1 material
(PP 6.1) Multivariate Gaussian - definition
Introduction to the multivariate Gaussian (or multivariate Normal) distribution.
From playlist Probability Theory
Probability: Definitions and Elementary Examples
This is the first video of a series from the Worldwide Center of Mathematics explaining the basics of probability. This video deals with some basic definitions and elementary probability examples. For more math videos, visit our channel or go to www.centerofmath.org.
From playlist Basics: Probability and Statistics
Probability Distribution Functions
We explore the idea of continuous probability density functions in a classical context, with a ball bouncing around in a box, as a preparation for the study of wavefunctions in quantum mechanics.
From playlist Quantum Mechanics Uploads
Alex SIMPSON - Probability sheaves
In [2], Tao observes that the probability theory concerns itself with properties that are \preserved with respect to extension of the underlying sample space", in much the same way that modern geometry concerns itself with properties that are invariant with respect to underlying symmetries
From playlist Topos à l'IHES
1. Probability Models and Axioms
MIT 6.041 Probabilistic Systems Analysis and Applied Probability, Fall 2010 View the complete course: http://ocw.mit.edu/6-041F10 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
Probability Theory - Part 5 - Product Probability Spaces [dark version]
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From playlist Probability Theory [dark version]
Introductory Probability Theory
A video introducing and deriving the foundations of probability theory up until the law of total probability and Bayes' theorem. This is an entry to the Summer of Math Exposition held by @3blue1brown. #SoME2 #3b1b #probability
From playlist Summer of Math Exposition 2 videos
Probability Theory - Part 5 - Product Probability Spaces
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From playlist Probability Theory
Markus Haase : Operators in ergodic theory - Lecture 1 : Operators dynamics versus ...
Abstract : The titles of the of the individual lectures are: 1. Operators dynamics versus base space dynamics 2. Dilations and joinings 3. Compact semigroups and splitting theorems Recording during the thematic meeting : "Probabilistic Aspects of Multiple Ergodic Averages " the December 6
From playlist Jean-Morlet Chair - Lemanczyk/Ferenczi
Amine Marrakchi: Ergodic theory of affine isometric actions on Hilbert spaces
The Gaussian functor associates to every orthogonal representation of a group G on a Hilbert space, a probability measure preserving action of G called a Gaussian action. This construction is a fundamental tool in ergodic theory and is the source of a large and interesting class of probabi
From playlist Probability and Statistics
A Friendly Introduction to Rigorous Probability Theory || Chapter 1, Probability Spaces
Here, I talk about why a rigorous (measure theoretic) framework for probability theory is needed, and also give an intuitive idea of various abstract ideas in rigorous probability such as sigma-algebras and the axioms of probability. This is my contribution to Grant Sanderson's (3blue1brow
From playlist Summer of Math Exposition Youtube Videos
MIT 6.042J Mathematics for Computer Science, Spring 2015 View the complete course: http://ocw.mit.edu/6-042JS15 Instructor: Albert R. Meyer License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.042J Mathematics for Computer Science, Spring 2015
Counting Outcomes (Size of Sample Space)
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From playlist Relative Frequency and Probability
The Poisson boundary: a qualitative theory (Lecture 2) by Vadim Kaimanovich
Program Probabilistic Methods in Negative Curvature ORGANIZERS: Riddhipratim Basu, Anish Ghosh and Mahan Mj DATE: 11 March 2019 to 22 March 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore The focal area of the program lies at the juncture of three areas: Probability theory o
From playlist Probabilistic Methods in Negative Curvature - 2019