the definition of random variable

Question

If we supposed that X is a random variable, is X - X a random variable? Could the outcome of an event is only 1? Cause X-X has only one outcome, and the possibility of it is 1. How about X + X?

Answer 1

A (real-valued) random variable $X$ is formally defined as a function $X : \Omega \to \mathbb R$, where $\Omega$ is the set of outcomes. So, $Y = X - X$ is also a function $\Omega \to \mathbb R$, specifically the zero function: $Y(\omega) = X(\omega) - X(\omega) = 0$ for all $\omega \in \Omega$.

Note that this is in fact a stronger statement than "$Y=0$ with probability one", which allows for the possibility that $Y(\omega) \neq 0$ on some nonempty subset of $\Omega$ which has probability zero.

One can generally without harm identify $X - X$ with the constant number zero, but technically it is a random variable.

To answer the other question, $Z = X+X$ is simply $2X$, which is of course also a random variable: formally, it is the function $Z : \Omega \to \mathbb R$ defined by $Z(\omega) = X(\omega) + X(\omega) = 2X(\omega)$ for all $\omega \in \Omega$.

Answer 2

Yes, $X-X$, and $X+X$ are random variables, assuming that $X$ is a random variable.

$X-X$, which is identically $0$, is considered a (discrete) random variable.