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 .