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I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be:

$p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and $|x-x_0|\le \epsilon$

But doesn't it mean an infinite PDF? For which distributions this could be true?

Lo Scrondo
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    That's going to zero, not actually zero. So these are different PDFs for each $\epsilon$. At any rate, a PDF can indeed be infinite or at least have an infinite limit at a point, such as $f(x)=\frac{x^{-1/2}}{2}$ on $(0,1]$ and zero elsewhere. – Ian Apr 27 '16 at 14:28

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There are indeed distributions - but no true functions - which can be interpreted as having infinite probability density at a point. The classical example is the Dirac delta function (which is not really a function).

Wouter
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  • Thanks for the answer. So you mean that for the Dirac delta, the $p(x)=\frac {1}{2\epsilon} \to \infty$ for $x=0$ and $p(x)=\frac {0}{2\epsilon}=0$ everywhere else? – Lo Scrondo Apr 27 '16 at 16:10
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    @LoScrondo Rigorously speaking, the Dirac delta is the limit (in an appropriate topology) of the uniform distributions on $(-\epsilon,\epsilon)$ as $\epsilon \to 0$. – Ian Apr 27 '16 at 20:00
  • Thank you @Ian..the fact is I could not figure to myself which distribution is described by $p(x)=\frac {1}{2\epsilon}$, with $|x-x_0|\le \epsilon$, in the limit $\epsilon \to 0$..doesn't it mean that at any point $x$ we get $p(x) \to \infty$? Or it's a family of distributions? – Lo Scrondo Apr 28 '16 at 04:40