Cross entropy of a random variable or a probability distribution function? I'm looking into the wikipedia page of cross entropy. https://en.wikipedia.org/wiki/Cross_entropy
$$H(p,q)=-\sum_{x\in \mathcal{X}} p(x)\log q(x)$$
It can be written as $$H(p,q) = H(p) + D_{KL} (p||q)$$ where $H(p)$ stands for the entropy of $p$ (which is a probability distribution function here), and $D_{KL} (p||q)$ means the Kullback–Leibler divergence between $p$ and $q$.
But when I looked into the definition of entropy. https://en.wikipedia.org/wiki/Entropy_(information_theory)
It says it is the entropy of a random variable.
$$H(x) = -\sum P(X_i)log(P(X_i))$$
So it really baffles me, is it the entropy of a probability distribution function or the entropy of a random variable? If it is the latter case, what if there are more than one probability distribution function of $x$? Like $p$ and $q$ on the cross entropy case. If so, it should be the entropy of a probability distribution?

Recall that a random variables $X$ is a measurable function $X:\Omega\to\mathbb R$ from a probability space $(\Omega, \mathcal F, p)$, where $\Omega$ is some set, $\mathcal F$ is a $\sigma$-algebra on $\Omega$, and $p$ is a probability measure for $(\Omega, \mathcal F)$. The distribution is just the "push forward" of $p$ to $\mathbb R$. Thus, a random variable can never have two different distributions.
The cross-entropy $H(p,q)$ is defined for two different random variables, one with the distribution $p$ and the other with the distribution $q$. Those two random variables just happen to have the same image, but since they create different distributions on this image, they are different functions.
The entropy is a property of a random variable that is computed from its distribution, so sometimes people talk about the entropy of a distribution.
