Editing Kronecker delta (section)

==Relationship to the Dirac delta function==
In [[probability theory]] and [[statistics]], the Kronecker delta and [[Dirac delta function]] can both be used to represent a [[discrete distribution]].  If the [[support (mathematics)|support]] of a distribution consists of points <math>\mathbf{x} = \{x_1,\cdots,x_n\}</math>, with corresponding probabilities <math>p_1,\cdots,p_n</math>, then the [[probability mass function]] <math>p(x)</math> of the distribution over <math>\mathbf{x}</math> can be written, using the Kronecker delta, as
<math display="block">p(x) = \sum_{i=1}^n p_i \delta_{x x_i}.</math>

Equivalently, the [[probability density function]] <math>f(x)</math> of the distribution can be written using the Dirac delta function as
<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function.  For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the [[Nyquist–Shannon sampling theorem]], the resulting discrete-time signal will be a Kronecker delta function.