Dirac Delta Function

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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 $\delta(x_o)$ means that the random variable can take the value $x_o$ 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 $\delta(x_o)$, where $x_o$ 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 $f(x)$, it gives the value of $f(x)$ at $x_o$, i.e. $f(x)*\delta(x_o) = f(x_o)$, where $*$ denotes convolution.