grandes-ecoles 2018 Q26

grandes-ecoles · France · centrale-maths2__pc Discrete Random Variables Existence of Expectation or Moments

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e. $$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$ For all $k \in \mathbb{N}$, give a necessary and sufficient condition on $x$ for $X^{k}$ to have a finite expectation. Express this expectation using $\zeta$.

Let $X$ be a random variable that follows the zeta distribution with parameter $x > 1$, i.e.
$$\forall n \in \mathbb{N}^{*}, \quad \mathbb{P}(X = n) = \frac{1}{\zeta(x) n^{x}}$$
For all $k \in \mathbb{N}$, give a necessary and sufficient condition on $x$ for $X^{k}$ to have a finite expectation. Express this expectation using $\zeta$.

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