grandes-ecoles 2018 Q32

grandes-ecoles · France · centrale-maths2__pc Discrete Probability Distributions Combinatorial Counting in Probabilistic Context

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).
For all $k \in \mathbb{N}^{*}$, show that $$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$

Let $U_{n}$ and $V_{n}$ be two independent random variables each following the uniform distribution on $\llbracket 1, n \rrbracket$. We denote by $W_{n} = U_{n} \wedge V_{n}$ (the GCD of $U_n$ and $V_n$).

For all $k \in \mathbb{N}^{*}$, show that
$$\mathbb{P}\left(W_{n} \in k\mathbb{N}^{*}\right) = \left(\frac{\lfloor n/k \rfloor}{n}\right)^{2}$$

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