How can a constant be a random variable? My professor posed us with the question, "Can “1” be a random variable?"
He said that we can argue both yes and no.
My argument for no is that 1 is a constant, and constants can not be a random variable. Also, there is no "randomness involved.
I was wondering if anyone could help me understand how the number 1 could be a random variable?
 A: Consider the function $f(x) = 7$. It's a pretty boring function! An input of $3$ gives an output of $7$, and an input of $-10$ gives an output of $7$, and so on.
Is it a function at all? Well, yes -- a function has one job, which is to take in inputs and return outputs. The fact that it's a boring function doesn't mean it's not a function, it just means that you're not going to get a Master's degree by carefully studying it.
In literally the same way, a random variable that's always $1$ is a perfectly good random variable. This is challenging, because the definition of a "random variable" is notoriously squishy in introductory probability contexts, and it's often not well-defined at all. The mathematically rigorous definition of it is that it's a measurable function from a particular sample space with a total measure of $1$, so the above analogy works quite well.
At an introductory level, we usually settle for something like "the outcome of a random process," or "a thing I can measure." Even with this looser definition, there's no actual requirement that the value changes upon repeated observations or anything like that; it's just a boring random variable.
A: It does fit the definition of random variable, so it is a random variable.
But more than that: in order for "random variable" to be a useful concept, it must include constants.  Otherwise you would have a very hard time doing algebra with random variables, e.g. if $X$ and $Y$ are random variables you couldn't talk about $X-Y$ as a random variable without checking that this is not constant.
