Editing Dirac delta function (section)

===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 absolutely continuous distributions).  For example, the probability density function {{math|''f''(''x'')}} of a discrete distribution consisting of points {{math|1='''x''' = {{brace|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}}}, with corresponding probabilities {{math|''p''<sub>1</sub>, ..., ''p<sub>n</sub>''}}, can be written as

<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

As another example, consider a distribution in 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

<math display="block">f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If {{math|1=''Y'' = g(''X'')}} is a continuous differentiable function, then the density of {{mvar|Y}} can be written as

<math display="block">f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx.  </math>

The delta function is also used in a completely different way to represent the [[local time (mathematics)|local time]] of a [[diffusion process]] (like [[Brownian motion]]).  The local time of a stochastic process {{math|''B''(''t'')}} is given by
<math display="block">\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math>
and represents the amount of time that the process spends at the point {{mvar|x}} in the range of the process.  More precisely, in one dimension this integral can be written
<math display="block">\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math>
where <math>\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}</math> is the [[indicator function]] of the interval <math>[x-\varepsilon,x+\varepsilon].</math>