Expected value of $X^TY$ in matrix form

Question

I want to compute $E(\mathbf{X}^T\mathbf{Y})$, where:

$$\mathbf{X} \sim N_d(\mathbf{\mu}, 4\mathbf{I}_d)$$ $$\mathbf{Y} \sim N_d(\mathbf{\mu}, 4\mathbf{I}_d)$$

Independence would seem to imply that this is the same as $E(\mathbf{X}^T)E(\mathbf{Y})$, which would be $||\mu||^2$.

Is this correct?

Answer 1

$\mu$ is a vector.

$$E(X^T)E(Y)=\sum_{i=1}^d \mu_i^2=\mu^T\mu=\|\mu\|^2$$