Dirac Delta Function - Applications To Probability Theory

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)