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Lecture

Probability and Measure: Fundamentals and Applications

In course

MSE-421: Statistical mechanics

This course presents an introduction to statistical mechanics geared towards materials scientists. The concepts of macroscopic thermodynamics will be related to a microscopic picture and a statistical interpretation. Lectures and exercises will be compleme ...

Description

This lecture introduces the fundamental concepts of probability theory and measure theory, covering topics such as events, probability axioms, joint probabilities, conditional probabilities, Bayes' formula, random variables, autocorrelation functions, characteristic functions, the central limit theorem, and the binomial distribution. The instructor explains how these concepts are applied in various scenarios, such as developing tests for rare diseases and predicting the distribution of independent variables. The lecture emphasizes the importance of understanding probability and measure theory in analyzing and interpreting data in different fields.

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Instructor

Michele Ceriotti

Michele Ceriotti received his Ph.D. in Physics from ETH Zürich in 2010. He spent three years in Oxford as a Junior Research Fellow at Merton College. Since 2013 he leads the laboratory for Computational Science and Modeling in the Institute of Materials at EPFL. His research revolves around the atomic-scale modelling of materials, based on the sampling of quantum and thermal fluctuations and on the use of machine learning to predict and rationalize structure-property relations. He has been awarded the IBM Research Forschungspreis in 2010, the Volker Heine Young Investigator Award in 2013, an ERC Starting Grant in 2016, and the IUPAP C10 Young Scientist Prize in 2018.

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Mathematics

Probability theory: Topics in probability theory

Related concepts (32)

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.

Conditional probability

In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(AB) or occasionally P_B(A).

Joint probability distribution

Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered for any given number of random variables. The joint distribution encodes the marginal distributions, i.e. the distributions of each of the individual random variables. It also encodes the conditional probability distributions, which deal with how the outputs of one random variable are distributed when given information on the outputs of the other random variable(s).

Probability theory

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.

Conditional probability distribution

In probability theory and statistics, given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter. When both and are categorical variables, a conditional probability table is typically used to represent the conditional probability.

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