Let $X \in \text{Random}(\mathbb{R}^{N \times 1})$ be a D5210: Random real column matrix such that

(i)	\begin{equation} \mathbb{E} |X|^2 < \infty \end{equation}

Let $a \in \mathbb{R}^{N \times 1}$ be a D5200: Real column matrix.

Denote $\mu : = \mathbb{E} X$. Using an add-and-substract trick with $- 2 \mu^T \mu + \mu^T \mu + \mu^T \mu = 0$, we have \begin{equation} \begin{split} \mathbb{E} |X - a|^2 & = \mathbb{E}[(X - a)^T (X - a)] \\ & = \mathbb{E}(X^T X - X^T a - a^T X + a^T a) \\ & = \mathbb{E}(X^T X - 2 X^T a + a^T a) \\ & = \mathbb{E}(X^T X) - 2 \mathbb{E} (X^T a) + a^T a \\ & = \mathbb{E}(X^T X) - 2 \mu^T a + a^T a \\ & = \mathbb{E}(X^T X) - 2 \mu^T \mu + \mu^T \mu + \mu^T \mu - 2 \mu^T a + a^T a \\ & = \mathbb{E}[(X - \mu)^T (X - \mu)] + \mu^T \mu - 2 \mu^T a + a^T a \\ & = \mathbb{E}[(X - \mu)^T (X - \mu)] + (\mu - a)^T (\mu - a) \\ & = \mathbb{E} |X - \mu|^2 + |\mu - a|^2 \\ \end{split} \end{equation} Subtracting $|\mu - a|^2$ from both sides now yields the result. $\square$