grandes-ecoles 2022 Q23

grandes-ecoles · France · centrale-maths1__pc Discrete Random Variables Covariance Matrix and Multivariate Expectation

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $U = \left(\begin{array}{c} u_1 \\ \vdots \\ u_n \end{array}\right)$ in $\mathcal{M}_{n,1}(\mathbb{R})$. We define the discrete random variable $X = U^\top Y$.
Show that $X$ admits a variance and that $$\mathbb{V}(X) = U^\top \Sigma_Y U.$$

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. Let $U = \left(\begin{array}{c} u_1 \\ \vdots \\ u_n \end{array}\right)$ in $\mathcal{M}_{n,1}(\mathbb{R})$. We define the discrete random variable $X = U^\top Y$.

Show that $X$ admits a variance and that
$$\mathbb{V}(X) = U^\top \Sigma_Y U.$$

Paper Questions

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