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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 $\mathcal{X}$.

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  • $\begingroup$ 1. Yes. 2. I don’t see why the subscripts here are necessary to begin with. I think you can just write p,q. $\endgroup$
    – Andrew
    Apr 4, 2023 at 18:02
  • $\begingroup$ 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. $\endgroup$ Apr 4, 2023 at 18:15

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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.

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