Is a constant a random variable?

Question

Can a constant $c$ be regarded as a random variable?

Is it correct to define it as a discrete random variable X with probability mass function $P(X=c)=1$?

Answer 1

A (real-valued) random variable is a function $X : \Omega \to \mathbb R$, where $\Omega$ is the sample space. Certainly given any constant $c$, we can define the associated constant function $X(\omega) = c$ for all $\omega \in \Omega$, and there's generally no confusion if we identify the number $c$ with the constant function. And indeed $X$ will have the probability mass function $P(X=c) = 1$.

The converse is not necessarily true, however. A random variable with probability mass function $P(X=c) = 1$ need not be constant. For example, if $\Omega$ is the unit interval $[0,1]$ with Lebesgue measure, we can define $$X(\omega) = \begin{cases} 1 & \text{if }\omega\text{ is irrational} \\ 0 & \text{if }\omega\text{ is rational} \\ \end{cases}$$ in which case $X$ has the probability mass function $P(X = 1) = 1$ but is not constant.

Answer 2

Yes; such a constant random variable has characteristic function $e^{ict}$, probability density $\delta (x-c)$ (a Dirac delta) etc.