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When a random variable $X$ has only one possible outcome $x_0$, the probability density function at $X=x_0$ is infinite, while the probability density at other locations is zero. Then the p.d.f is exactly a delta function $\Pr(X=x) = \delta(x=x_0)$.

However, when I tried to calculate the entropy of the random variable, the problem arises. How can I calculate the integral $\int_{-\infty}^{+\infty}{\delta(x-x_0)\log\delta(x-x_0) \, dx}$?

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    Mathematicians prefer to use measures rather than "densities" for probability distributions that are not absolutely continuous. $Pr(X=x) = 1 \ne \infty = \delta(x-x_0)$2012-09-04
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    @Strin : I'd write "probability density function" in the title. "Distribution function" is usually construed as "cumulative distribution function".2012-09-04

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