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

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 $$\infty$$, $$-\infty$$?

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

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