Editing Probability distribution (section)

===Dirac delta representation===
A discrete probability distribution is often represented with [[Dirac measure]]s, also called one-point distributions (see below), the probability distributions of [[Degenerate distribution|deterministic random variable]]s. For any outcome <math>\omega</math>, let <math>\delta_\omega</math> be the Dirac measure concentrated at <math>\omega</math>. Given a discrete probability distribution, there is a countable set <math>A</math> with <math>P(X \in A) = 1</math> and a probability mass function <math>p</math>. If <math>E</math> is any event, then
<math display="block">P(X \in E) = \sum_{\omega \in A} p(\omega) \delta_\omega(E),</math>
or in short,
<math display="block">P_X = \sum_{\omega \in A} p(\omega) \delta_\omega.</math>

Similarly, discrete distributions can be represented with the [[Dirac delta function]] as a [[Generalized function|generalized]] [[probability density function]] <math>f</math>, where
<math display="block">f(x) = \sum_{\omega \in A} p(\omega) \delta(x - \omega),</math>
which means
<math display="block">P(X \in E) = \int_E f(x) \, dx = \sum_{\omega \in A} p(\omega) \int_E \delta(x - \omega) = \sum_{\omega \in A \cap E} p(\omega)</math>
for any event <math>E.</math><ref>{{Cite journal|last=Khuri|first=André I.|date=March 2004| title=Applications of Dirac's delta function in statistics|journal=International Journal of Mathematical Education in Science and Technology| language=en|volume=35|issue=2|pages=185–195| doi=10.1080/00207390310001638313|s2cid=122501973|issn=0020-739X}}</ref>