Dirac Delta Function

From Computational Statistics (CSE383M and CS395T)
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Massed Distribution

Dirac delta distribution is a special case of continuous probability distribution. It can be thought of as the limiting form of Normal distribution of a continuous random variable. The distribution <math>\delta(x_o)</math> means that the random variable can take the value <math>x_o</math> with probability 1 and other values with probability 0.

When the variance of a normal random variable approaches zero, the probability density function of this variable becomes less spread out along the x-axis but becomes increasingly taller in such a way that the area under the curve always remains 1. In the limit when the variance approaches zero, the pdf becomes a dirac delta function , denoted as <math>\delta(x_o)</math>, where <math>x_o</math> is the value at which the distribution is centered.

Dirac delta function has the nice property that when this function is convolved with another function <math>f(x)</math>, it gives the value of <math>f(x)</math> at <math>x_o</math>, i.e. <math>f(x)*\delta(x_o) = f(x_o)</math>, where <math>*</math> denotes convolution.