Expected squared euclidean distance

Hi,

can someone help me with this problem?

Quote:

Let $\displaystyle X, Y$ be two independent $\displaystyle d$-dimensional random variables with Gaussian distribution $\displaystyle N(\mu_{X},\sigma^2 \mathbb{I}), N(\mu_{Y},\sigma^2 \mathbb{I})$, where $\displaystyle \mathbb{I}$ is the identity matrix in $\displaystyle \mathbb{R}^d, \sigma \in \mathbb{R}$ with $\displaystyle \sigma > 0$, and $\displaystyle \mu_X , \mu_Y \in \mathbb{R}^d$.

Compute the expected squared distance $\displaystyle E(||X-Y||^2)$.

Hint: Use that $\displaystyle ||X-Y||^2 = ||X||^2 + ||Y||^2 - 2<X,Y>$

This is what I've got so far:

$\displaystyle E(||X-Y||^2) = $

$\displaystyle E(||X||^2 + ||Y||^2 - 2<X,Y>) = $

$\displaystyle E(||X||^2) + E(||Y||^2) - 2E(<X,Y>) = $

$\displaystyle E(\sum_{i = 1}^d X_i^2) + E(\sum_{i = 1}^d Y_i^2) - 2E(\sum_{i=1}^d X_i Y_i) = $

$\displaystyle E(\sum_{i = 1}^d X_i^2) + E(\sum_{i = 1}^d Y_i^2) - 2 \sum_{i = 1}^d E(X_i Y_i) = $

$\displaystyle E(\sum_{i = 1}^d X_i^2) + E(\sum_{i = 1}^d Y_i^2) - 2 \sum_{i = 1}^d E(X_i) E(Y_i) = $

$\displaystyle E(\sum_{i = 1}^d X_i^2) + E(\sum_{i = 1}^d Y_i^2) - 2 \sum_{i = 1}^d \mu_{X_i} \mu_{Y_i} = $

$\displaystyle E(\sum_{i = 1}^d X_i^2) + E(\sum_{i = 1}^d Y_i^2) - 2 <\mu_X , \mu_Y> $

Now, what could be done further (especially regarding the 2 first addends)?

Thank you very much!