Random variable notation when writing down probability distributions

I want to discuss discrete random variables that take on outcomes from some finite set. Is there a sense in which every probability distribution is associated with a random variable and vice versa?

1. Is it true that if I have some finite set $$\mathcal{X}$$, then a random variable $$\mathsf{X}$$ which takes on values in this set always has some underlying probability distribution $$p_{\mathsf{X}}$$?

2. Next, is it true that if I want to speak about two distinct probability distributions $$p$$ and $$q$$, then I should not use the notation $$p_{\mathsf{X}}$$ and $$q_{\mathsf{X}}$$? My intial feeling is that I should use $$p_{\mathsf{X}}$$ and $$q_{\mathsf{X'}}$$ where $$\mathsf{X}, \mathsf{X'}$$ are random variables which both take on values in $$\mathcal{X}$$.

• 1. Yes. 2. I don’t see why the subscripts here are necessary to begin with. I think you can just write p,q. Apr 4, 2023 at 18:02
• In the paper I am reading, they use subscripts so I just wanted to understand what the correct notation would be, if one were to do so. Apr 4, 2023 at 18:15

For 1, the answer is yes - if $$X$$ takes on random values from $$\mathcal{X}$$ then there is implicity a distribution $$p_X(x) = P(X = x)$$.
For 2, notation is generally a matter of preference. However, I would say that typically you would either say that random variables $$X$$ and $$X'$$ have distributions $$p_X$$ and $$p_{X'}$$, or you would say they have distributions $$p$$ and $$q$$ respectively. As long as it's clear which function gives the distribution for which variable, it's fine.