grandes-ecoles 2019 Q38

grandes-ecoles · France · centrale-maths2__pc Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. For every $i \in \llbracket 0, n \rrbracket$, show $\mathbb{P}(M_n > i) = p_i$.

For every integer $n \geqslant 2$, equip the set $\Omega_n$ of permutations of $\llbracket 1, n \rrbracket$ with the uniform probability. Let $p_i$ denote the probability that a permutation is alternating up (with $p_0 = p_1 = 1$). Define the random variable $M_n$ on $\Omega_n$ by: $M_n(\sigma) = k+1$ where $k$ is the largest integer such that $(\sigma(1), \ldots, \sigma(k))$ is alternating up. For every $i \in \llbracket 0, n \rrbracket$, show $\mathbb{P}(M_n > i) = p_i$.

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