grandes-ecoles 2018 Q29

grandes-ecoles · France · centrale-maths2__pc Discrete Probability Distributions Proof of Distributional Properties or Symmetry

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.

Let $x$ be a real number such that $x > 1$ and let $X$ be a random variable that follows the zeta distribution with parameter $x$, i.e.
$$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$
Let $(q_{1}, \ldots, q_{n}) \in \mathcal{P}^{n}$ be an $n$-tuple of distinct prime numbers. Show that the events $(X \in q_{1}\mathbb{N}^{*}), \ldots, (X \in q_{n}\mathbb{N}^{*})$ are mutually independent.

Paper Questions

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