We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ defined on $(\Omega, \mathcal{B}, \mathbb{P})$ with real values and define the random vector $Y(\omega) = \left(\begin{array}{c} Y_1(\omega) \\ \vdots \\ Y_n(\omega) \end{array}\right)$. The covariance matrix $\Sigma_Y$ has general term $\sigma_{i,j} = \operatorname{cov}(Y_i, Y_j)$. Verify that $\Sigma_Y$ is a symmetric matrix, that $$\Sigma_Y = \mathbb{E}\left((Y - \mathbb{E}(Y))(Y - \mathbb{E}(Y))^\top\right)$$ and that, if $U$ is a constant vector in $\mathcal{M}_{n,1}(\mathbb{R})$, then $$\Sigma_{Y+U} = \Sigma_Y.$$
We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ defined on $(\Omega, \mathcal{B}, \mathbb{P})$ with real values and define the random vector $Y(\omega) = \left(\begin{array}{c} Y_1(\omega) \\ \vdots \\ Y_n(\omega) \end{array}\right)$. The covariance matrix $\Sigma_Y$ has general term $\sigma_{i,j} = \operatorname{cov}(Y_i, Y_j)$.
Verify that $\Sigma_Y$ is a symmetric matrix, that
$$\Sigma_Y = \mathbb{E}\left((Y - \mathbb{E}(Y))(Y - \mathbb{E}(Y))^\top\right)$$
and that, if $U$ is a constant vector in $\mathcal{M}_{n,1}(\mathbb{R})$, then
$$\Sigma_{Y+U} = \Sigma_Y.$$