We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $p \in \mathbb{N}^*$ and $M \in \mathcal{M}_{p,n}(\mathbb{R})$. We define the discrete random variable $Z = MY$, with values in $\mathcal{M}_{p,1}(\mathbb{R})$. Justify that $Z$ admits an expectation and express $\mathbb{E}(Z)$ in terms of $\mathbb{E}(Y)$. Show that $Z$ admits a covariance matrix $\Sigma_Z$ and that
$$\Sigma_Z = M \Sigma_Y M^\top.$$