grandes-ecoles 2022 Q24

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 $\Sigma_Y$. The objective is to show that $$\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1.$$ We denote by $r$ the rank of the covariance matrix of $Y$.
Handle the case where $r = n$.

We consider $n$ discrete random variables $Y_1, \ldots, Y_n$ with random vector $Y$ and covariance matrix $\Sigma_Y$. The objective is to show that
$$\mathbb{P}\left(Y - \mathbb{E}(Y) \in \operatorname{Im}\Sigma_Y\right) = 1.$$
We denote by $r$ the rank of the covariance matrix of $Y$.

Handle the case where $r = n$.

Paper Questions

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