grandes-ecoles 2018 Q38

grandes-ecoles · France · centrale-maths1__official Continuous Probability Distributions and Random Variables Distribution of Transformed or Combined Random Variables

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$. Justify that $g(X)$ admits at least one median. One may consider the function $G$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for every real number $t$, $G(t) = \mathbb{P}(g(X) \leqslant t)$, and examine the set $G^{-1}([1/2, 1])$.

We consider $g(X)$ where $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ is a random variable with independent Rademacher coefficients and $g(M) = \|M \cdot u\|$ for a fixed unit vector $u$. Justify that $g(X)$ admits at least one median. One may consider the function $G$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for every real number $t$, $G(t) = \mathbb{P}(g(X) \leqslant t)$, and examine the set $G^{-1}([1/2, 1])$.

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