Expected value of outer product of multivariate normal random vector with itself

Question

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?

Answer 1

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, $\mathbb{E}\left[tt^{\intercal}\right]=\Psi+\mu\mu^{\intercal}$.