**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:

“Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical *probability* provides loopholes, and discrepancies are expunged by Faith.”

—Ursula K. Le Guin (b. 1929)

“The whole *theory* of modern education is radically unsound. Fortunately in England, at any rate, education produces no effect whatsoever. If it did, it would prove a serious danger to the upper classes, and probably lead to acts of violence in Grosvenor Square.”

—Oscar Wilde (1854–1900)