Smooth functions | Articles containing proofs | Functional analysis | Generalizations of the derivative | Generalized functions
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function. A function is normally thought of as acting on the points in the function domain by "sending" a point in its domain to the point Instead of acting on points, distribution theory reinterprets functions such as as acting on test functions in a certain way. In applications to physics and engineering, test functions are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset . (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by or Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that "acts on" a test function by "sending" it to the number which is often denoted by This new action of is a scalar-valued map, denoted by whose domain is the space of test functions This functional turns out to have the two defining properties of what is known as a distribution on : it is linear and also continuous when is given a certain topology called the canonical LF topology. The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like that arise from functions in this way are prototypical examples of distributions, but many cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures. It is nonetheless still possible to down to a simpler family of related distributions that do arise via such actions of integration. More generally, a distribution on is by definition a linear functional on that is continuous when is given a topology called the canonical LF topology. This leads to the space of (all) distributions on , usually denoted by (note the prime), which by definition is the space of all distributions on (that is, it is the continuous dual space of ); it is these distributions that are the main focus of this article. Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples. (Wikipedia).
What is a Sampling Distribution?
Intro to sampling distributions. What is a sampling distribution? What is the mean of the sampling distribution of the mean? Check out my e-book, Sampling in Statistics, which covers everything you need to know to find samples with more than 20 different techniques: https://prof-essa.creat
From playlist Probability Distributions
The Normal Distribution (1 of 3: Introductory definition)
More resources available at www.misterwootube.com
From playlist The Normal Distribution
Statistics: Introduction to the Shape of a Distribution of a Variable
This video introduces some of the more common shapes of distributions http://mathispower4u.com
From playlist Statistics: Describing Data
Distributions - Statistical Inference
In this video I talk about distribution, how to visualize it and also provide a concrete definition for it.
From playlist Statistical Inference
Uniform Probability Distribution Examples
Overview and definition of a uniform probability distribution. Worked examples of how to find probabilities.
From playlist Probability Distributions
(PP 6.1) Multivariate Gaussian - definition
Introduction to the multivariate Gaussian (or multivariate Normal) distribution.
From playlist Probability Theory
Discrete Math - 7.2.2 Random Variables and the Binomial Distribution
Introduction to random variables and finding probability of an event or cumulative probability using the binomial distribution. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz
From playlist Discrete Math I (Entire Course)
Sampling Distribution of the PROPORTION: Friends of P (12-2)
The sampling distribution of the proportion is the probability distribution of all possible values of the sample proportions. It is analogous to the Distribution of Sample Means. When the sample size is large enough, the sampling distribution of the proportion can be approximated by a norm
From playlist Sampling Distributions in Statistics (WK 12 - QBA 237)
Elchanan Mossel - Some mathematical theorems on agreement and learning in networks - IPAM at UCLA
Recorded 16 February 2022. Elchanan Mossel of the Massachusetts Institute of Technology presents "Some mathematical theorems on agreement and learning in networks" at IPAM's Mathematics of Collective Intelligence Workshop. Learn more online at: http://www.ipam.ucla.edu/programs/workshops/m
From playlist Workshop: Mathematics of Collective Intelligence - Feb. 15 - 19, 2022.
PUBLIC OPENING featuring Cédric Villani: The Many Facets of Entropy [2014]
Video taken from: http://www.fields.utoronto.ca/programs/scientific/fieldsmedalsym/14-15/
From playlist Mathematics
Clojure Conj 2012 - Clojure Data Science
Clojure Data Science by: Edmund Jackson Data science / big data exists at the overlap of traditional analytics and large scale computation. As such, neither the traditional tools of analytics (R, Mathematica, Matlab) nor mainstreams languages (Java, C++, C#) supply its requirements well a
From playlist Clojure Conf 2012
Probability Distribution Functions - EXPLAINED!
Probability distribution functions are functions that map an event to the probability of occurrence of that event. Let's talk about them. For more information, check out the blog post on probability fundamentals in Machine Learning: https://towardsdatascience.com/probability-for-machine-l
From playlist The Math You Should Know
Tilmann Gneiting: Isotonic Distributional Regression (IDR) - Leveraging Monotonicity, Uniquely So!
CIRM VIRTUAL EVENT Recorded during the meeting "Mathematical Methods of Modern Statistics 2" the June 02, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians
From playlist Virtual Conference
Dependence Uncertainty and Risk - Prof. Paul Embrechts
Abstract I will frame this talk in the context of what I refer to as the First and Second Fundamental Theorem of Quantitative Risk Management (1&2-FTQRM). An alternative subtitle for 1-FTQRM would be "Mathematical Utopia", for 2-FTQRM it would be "Wall Street Reality". I will mainly conce
From playlist Uncertainty and Risk
Hierarchical Modeling of High-dimensional Human Immuno-phenotypic Diversity by Saumyadipta Pyne
DISCUSSION MEETING : MATHEMATICAL AND STATISTICAL EXPLORATIONS IN DISEASE MODELLING AND PUBLIC HEALTH ORGANIZERS : Nagasuma Chandra, Martin Lopez-Garcia, Carmen Molina-Paris and Saumyadipta Pyne DATE & TIME : 01 July 2019 to 11 July 2019 VENUE : Madhava Lecture Hall, ICTS, Bangalore
From playlist Mathematical and statistical explorations in disease modelling and public health
Online-Vortrag "Blick in den Körper: Über das Inverse und medizinische Bildgebung" (Director's Cut)
Aufzeichnung (Director's Cut): Prof. Dr. Benedikt Wirth erläutert im Rahmen der öffentlichen Reihe "Brücken in der Mathematik" die mathematischen Konzepte hinter der medizinischen Bildgebung. Darum geht es: Moderne Technik erlaubt den Blick in den Körper, ohne ihn zu öffnen. Es wird sozu
From playlist Brücken in der Mathematik
Learnability can be undecidable | AISC
For slides and more information on the paper, visit https://aisc.ai.science/events/2019-06-17/ Discussion lead: Abdulrahman Al-lahham Discussion facilitator: Mehdi Garrousian
From playlist Math and Foundations
Identifying, symmetric, skewed, uniform, and bell-shaped distributions
From playlist Unit 1: Descriptive Statistics