TITLE: Expected value of outer product of multivariate normal random vector with itself
QUESTION [6 upvotes]: Let's say I have a random vector $\boldsymbol{t}$ that is distributed according to a multivariate normal distribution:
$$
\boldsymbol{t} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Psi})
$$
I now want to find the expected value of the outer product of this random vector:
$$
\mathbb{E}\left[\boldsymbol{t}\boldsymbol{t}^\intercal\right]
$$
Is there a closed-form solution to this problem? In my studies, I have stumbled across the Wishart distribution. Might this be a way to tackle this problem?

REPLY [6 votes]: First, note that
$$
\left(t-\mu\right)\left(t-\mu\right)^{\intercal}=tt^{\intercal}-t\mu^{\intercal}-\mu t^{\intercal}+\mu\mu^{\intercal}.
$$
Therefore,
\begin{align*}
\operatorname{Var}(t) & =\mathbb{E}\left[\left(t-\mu\right)\left(t-\mu\right)^{\intercal}\right]\\
 & =\mathbb{E}\left[tt^{\intercal}\right]-\mathbb{E}\left[t\right]\mu^{\intercal}-\mu\mathbb{E}\left[t^{\intercal}\right]+\mu\mu^{\intercal}\\
 & =\mathbb{E}\left[tt^{\intercal}\right]-\mu\mu^{\intercal}.
\end{align*}
You already know that $\operatorname{Var}(t)=\Psi$.
Moving some terms around, $\mathbb{E}\left[tt^{\intercal}\right]=\Psi+\mu\mu^{\intercal}$.