The difference between standard deviation $=0$ and no standard deviation

Question

I would like to ask if there is the difference in the interpretation standard deviation of the Gaussian $=0$, and there is no standard deviation for the Gaussian.

Is it the same? Or there is some difference!

For example, when we build the Gaussian for background, and the pixsel is coming (which always has the same value, and it doesn't affect by any noise - let's say perfect conditions), then gaussian built will have the same mean value, and std=0 or if there is (in general) the definition in math: gaussian without any std at all? Is it possible?

Thank you!

Answer 1

A constant variable's standard deviation is well-defined and equal to $0$. A Cauchy distribution's standard deviation is undefined, and therefore can't be described as $0$ or any other number; indeed, "a Cauchy distribution's standard deviation" doesn't exist.

Answer 2

A Gaussian distribution with mean $\mu$ and variance $\sigma^2$ is typically defined as a continuous distribution with probability density function $$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x - \mu)^2/(2\sigma^2)}. $$

If you set $\sigma=0$ then the density function becomes undefined. Taking a limit as $\sigma \to 0^+,$ the density function goes to $0$ when $x\neq \mu$ and goes to infinity when $x = \mu.$

You can consider a known value (with no uncertainty) to be a discrete random variable that takes a value $\mu$ with probability $1.$ The variance of this variable is zero.

So you certainly can assign random variables to every pixel such that some of the variables are Gaussian with positive variance and some of the variables have zero variance. I would not call the zero-variance variables "Gaussian," however.