Solved – Expectation with subscript

The subscript just means that the expectation is with respect to the joint distribution of $(A,B)$. Use of the subscript does not add anything in this case, since there is nothing in the expectation that depends on the random variable $B$. In this case you have:

$$begin{equation} begin{aligned} mathbb{E}_{A,B}(A^2) &= int int a^2 p(a,b) da text{ } db \[6pt] &= int int a^2 p(a) p(b|a) da text{ } db \[6pt] &= int a^2 p(a) Big( int p(b|a) db Big) da \[6pt] &= int a^2 p(a) da \[6pt] &= mathbb{E}_A(A^2). \[6pt] end{aligned} end{equation}$$

In this kind of case you really don't need any subscript on the expectation operator. In the absence of a subscript we just assume that the expectation is taken over the joint distribution of all random variables in the expression.