Expected value is linear, so if $X$ is the only random quantity in the expression
$E[((XB) - E(XB))^T R] = 0$.
EDIT: with the question changed to make $R$ random as well, the objective becomes
$E(B^T X^T R) - E(XB)^T E(R) = B^T C = C^T B$ where $C = E(X^T R) - E(X^T) E(R)$, while the
constraint is $E(X) B = 1$ (I assume that's a vector of all $1$'s?).
Case 1: If $1$ is not in the column space of the matrix $E(X)$, the problem is infeasible: there are
no solutions to the constraint.
Case 2: If $1$ is in the column space of $E(X)$ and $C^T$ is in the row space of $E(X)$, the objective is constant on the solution space.
Any solution of the constraint maximizes it.
Case 3: If $1$ is in the column space of $E(X)$ and $C^T$ is not in the row space of $E(X)$, the objective is unbounded. You can take a vector $V$ such that $E(X) V = 0$
and $C^T V > 0$, and by adding a suitable multiple of $V$ to any solution of $E(X) B = 1$ you get a solution with $C^T X$ arbitrarily large.