Variance of constant function and dirac delta

Question

I have seen claims saying constant function has 'null variance' (i.e., $0$). I find this conflicting with the fact that dirac delta also has variance of $0$ (since it is the extreme case of a very concentrated Gaussian), since they are more of 2 species at the opposite ends of a spectrum. I'm thinking maybe the problem lies in the definition of the variance of a function and and distribution, but not very sure.

Answer 1

I think you are confusing a constant random variable with a constant probability density function.

1. If $X = x_0$, then it means that the random variable $X$ can take one single value named $x_0$, hence $X\sim\mathcal{Delta}(x_0)$ and its PDF is given by $f_X(x) = \delta(x-x_0)$, which implies a null variance. In fact, it's the way to describe a deterministic (i.e. non-random) variable in the probabilistic language.

2. The opposite end of spectrum, as you said, corresponds to the case where all events are of equal probability, hence a constant probability density function $f_X(x) = c$; thus, the considered random variable has to be uniformly distributed, i.e. $X\sim\mathcal{Unif}([0,c])$ for instance, and it has a non-zero variance.