grandes-ecoles 2018 Q34

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

Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.
Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.

Let $(\ell_k)_{k \in \mathbb{N}^*}$ be the limits defined in Q33, where $\ell_k = \lim_{n \to \infty} \mathbb{P}(W_n = k)$ and $W_n = U_n \wedge V_n$ for independent uniform random variables $U_n, V_n$ on $\llbracket 1, n \rrbracket$.

Using the result of Q33, deduce that $(\ell_{k})_{k \in \mathbb{N}^{*}}$ defines a probability distribution on $\mathbb{N}^{*}$.

Paper Questions

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