Applications To Probability Theory
In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent fully continuous distributions). For example, the probability density function ƒ(x) of a discrete distribution consisting of points, with corresponding probabilities, can be written as
As another example, consider a distribution which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as
The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process B(t) is given by
and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written
where is the indicator function of the interval .
Read more about this topic: Dirac Delta Function
Famous quotes containing the words probability and/or theory:
“Liberty is a blessing so inestimable, that, wherever there appears any probability of recovering it, a nation may willingly run many hazards, and ought not even to repine at the greatest effusion of blood or dissipation of treasure.”
—David Hume (1711–1776)
“... the first reason for psychology’s failure to understand what people are and how they act, is that clinicians and psychiatrists, who are generally the theoreticians on these matters, have essentially made up myths without any evidence to support them; the second reason for psychology’s failure is that personality theory has looked for inner traits when it should have been looking for social context.”
—Naomi Weisstein (b. 1939)