**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Independence (probability theory)

Summary

Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.
When dealing with collections of more than two events, two notions of independence need to be distinguished. The events are called pairwise independent if any two events in the collection are independent of each other, while mutual independence (or collective independence) of events means, informally speaking, that each event is independent of any combination of other events in the collection. A similar notion exists for collections of random variables. Mutual independence implies pairwise independence, but not the other way around. In the standard literature of probability theory, statistics, and stochastic processes, independence without further qualification usually refers to mutual independence.
Two events and are independent (often written as or , where the latter symbol often is also used for conditional independence) if and only if their joint probability equals the product of their probabilities:
indicates that two independent events and have common elements in their sample space so that they are not mutually exclusive (mutually exclusive iff ). Why this defines independence is made clear by rewriting with conditional probabilities as the probability at which the event occurs provided that the event has or is assumed to have occurred:
and similarly
Thus, the occurrence of does not affect the probability of , and vice versa. In other words, and are independent to each other. Although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if or are 0.

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (18)

Related publications (61)

Related people (13)

Related concepts (22)

Related lectures (98)

Ontological neighbourhood

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).

Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely 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 be equal to that sample.

A basic course in probability and statistics

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a

Introduction à la physique des plasmas destinée à donner une vue globale des propriétés essentielles et uniques d'un plasma et à présenter les approches couramment utilisées pour modéliser son comport

Plasma Physics: Collisions and Resistivity

Covers Coulomb collisions and resistivity in plasma, highlighting their random walk nature.

Probability Theory: Independence and Conditional Independence

Explores independence and conditional independence in probability theory through mathematical definitions and examples.

Probability: Independence

Explores the concept of independence in probability theory, showing how events can occur without influencing each other.

The field of synthetic data is more and more present in our everyday life. The transportation domain is particularly interested in improving the methods for the generation of synthetic data in order to address the privacy and availability issue of real dat ...

2023,

This paper presents explicit solutions for two related non-convex information extremization problems due to Gray and Wyner in the Gaussian case. The first problem is the Gray-Wyner network subject to a sum-rate constraint on the two private links. Here, ou ...

The quantification of uncertainties can be particularly challenging for problems requiring long-time integration as the structure of the random solution might considerably change over time. In this respect, dynamical low-rank approximation (DLRA) is very a ...