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I have recently come across the notation $p(X)$ i.e here. I have not seen this notation of mixing small $p$ and a random variable $X$ before. For instance lets assume we have a fair die such that $p(x) = 1/6$ for all $x$ in the sample space {1,2,3,4,5,6}. We draw a random variable $X \sim p(x)$. What would $p(X)$ denote in this case?

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$p$ either denotes the PMF or PDF of a distribution. It is a non-random function.

You can plug a random variable into a function to get a new random variable.

When discussing entropy in the context of information theory, one often plugs in a random variable $X$ into its own PDF/PMF $p$, yielding a new random variable $p(X)$.

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  • $\begingroup$ So for the example in the questions p(X) is a random variable but the outcome is 1/6 always? $\endgroup$
    – sn3jd3r
    Apr 12, 2020 at 2:13
  • $\begingroup$ @sn3jd3r Yes. ${}$ $\endgroup$
    – angryavian
    Apr 12, 2020 at 2:16

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