Definition of cross entropy (Wiki link for details): $$H(p,q) = H(p) + \mathcal{D}_{KL}(p||q)$$
Definition of joint entropy: \begin{align*} H(X,Y) &= -\sum_x \sum_y p(x,y) \log p(x,y)\\ &= H(X) + H(Y|X) \end{align*}
What are the differences between these two? I'm having difficult to distinguish the concept of entropy for random variables versus the concept of entropy for distributions?
Are there any connections between these two? E.g., can we use joint entropy to prove cross entropy formula, if not, how would you derive cross-entropy formula?