The probability of two independent random variables?

Question

Let $X, X'$ be independent with $X \sim p(x)$, $X' \sim r(x)$ for $x, x' \in X$.

I don't understand this equation:

$\sum p(x)r(x)=Pr(X=X')$

What is intuitive to me is if $X \sim p(x)$, $X' \sim p(x)$ for $x, x' \in X$, then $\sum p(x)p(x) = Pr(X=X')$.

Could anyone please explain a bit about why $\sum p(x)r(x)=Pr(X=X')$ above? Does it mean that $\sum p(x)r(x) = \sum p(x) p(x)=Pr(X=X')$ ?

Source: http://staff.ustc.edu.cn/~cgong821/Wiley.Interscience.Elements.of.Information.Theory.Jul.2006.eBook-DDU.pdf#page=67

Answer 1

If $X,X'$ are discrete random variables that are independent then: $$\Pr(X=X')=\sum_x\Pr(X=x=X')=\sum_x\Pr(X=x)\Pr(X'=x)$$where $x$ ranges over a countable set that serves as common support for $X$ and $X'$.

Let me know if things are still not clear.

Answer 2

One way is to use the following:

$$ \Pr(X=X') = \sum_{x} \Pr(X=X'\,\vert \, X'=x)\Pr(X'=x)$$

and the right hand side is:

$$ \sum_{x} \Pr(X=x\,\vert \, X'=x)\Pr(X'=x) = \sum_{x} \Pr(X=x)\Pr(X'=x)$$