In the context of Normal distribution, could $\mu$ and $\sigma$ be any value, such as $10^{-99}, 10^{-100}$, or even $\infty$, $-\infty$?

Question

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 $\infty$, $-\infty$?

Answer 1

$\mu$ 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.