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Lecture# Probability Theory: Independence and Conditional Independence

Description

This lecture covers the concepts of independence and conditional independence in probability theory. It explains how events A and B are considered independent when the occurrence of one does not affect the other. The lecture also delves into the mathematical definitions and examples of independence, including mutual independence and pairwise independence. Additionally, it explores the concept of conditional independence given another event. Various scenarios and calculations are presented to illustrate these concepts.

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MATH-233: Probability and statistics

The course gives an introduction to probability and statistics for physicists.

Year

A year is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the hours of daylight, and, consequently, vegetation and soil fertility. In temperate and subpolar regions around the planet, four seasons are generally recognized: spring, summer, autumn and winter.

Tropical year

A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice. It is the type of year used by tropical solar calendars. The solar year is one type of astronomical year and particular orbital period.

Independence (probability theory)

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.

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.

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.