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this is the PDF of a normal distribution.

${\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}$

where

  • $\mu$ is the mean or expectation of the distribution (and also its median and mode)
  • $\sigma$ is the standard deviation

Is there a notion like the domain of definition of a function to specify a range where $\mu$ and $\sigma$ take on.

In another word, could $\mu$ and $\sigma$ be any value, such as $10^{-99}, 10^{-100}$, or even $\infty$, $-\infty$?

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  • $\begingroup$ $\mu$ can be any real number and $\sigma$ can be any positive real number. $\endgroup$
    – Qiaochu Yuan
    Oct 5, 2019 at 4:17

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$\mu$ can be any real number and $\sigma$ can be any positive real number. They cannot be $\pm \infty$ as those are not numbers and we cannot plug them into the formula. There is nothing special about $\mu$ having a tiny value. If $\sigma$ is tiny the value of the random variable is forced to be very close to the mean.

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  • $\begingroup$ In the sense of distributions, $\sigma$ can be $0$. You get the degenerate distribution. $\endgroup$
    – eyeballfrog
    Oct 5, 2019 at 4:22
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    $\begingroup$ Yes but the OP is talking about it as a PDF over real numbers, so here $\sigma >0$. @eyeballfrog $\endgroup$
    – nicomezi
    Oct 5, 2019 at 4:29

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