Distribution (mathematics)

:This page deals with generalized functions in mathematical analysis. It is not about probability distributions.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. They extend the concept of derivative to all continuous functions and beyond and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution.

Related Topics:
Mathematical analysis - Function - Probability distribution - Derivative - Continuous - Partial differential equation - Physics - Engineering - Dirac delta

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

"Generalized functions" were introduced by Sergei Sobolev in 1935. They were independently discovered in late 1940s by Laurent Schwartz, who developed a comprehensive theory of distributions.

Related Topics:
Sergei Sobolev - Laurent Schwartz

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Sometimes, people talk of a "probability distribution" when they just mean "probability measure", especially if it is obtained by taking the product of the Lebesgue measure by a positive, real-valued measurable function of integral equal to 1.

Related Topics:
Probability - Measure - Lebesgue measure

~ ~ ~ ~ ~ ~ ~ ~ ~ ~