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,\dots,x_n\}</math>, with corresponding probabilities <math>p_1,\dots,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
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.
