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Omni-MATH / The spikiness of a sequence a₁,a₂,…,a_(n) of at least two real numbers is the sum ∑_(i=1)^(n−1)|a_(i+1)−a_(i)|. Suppose x₁,x₂,…,x₉ are chosen uniformly and randomly from the interv…

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problem

The spikiness of a sequence a1,a2,…,ana_{1}, a_{2}, \ldots, a_{n} of at least two real numbers is the sum ∑i=1n−1|ai+1−ai|\sum_{i=1}^{n-1}\left|a_{i+1}-a_{i}\right|. Suppose x1,x2,…,x9x_{1}, x_{2}, \ldots, x_{9} are chosen uniformly and randomly from the interval [0,1][0,1]. Let MM be the largest possible value of the spikiness of a permutation of x1,x2,…,x9x_{1}, x_{2}, \ldots, x_{9}. Compute the expected value of MM.
Plain-text mathematical notation (without MathML)
The spikiness of a sequence a₁,a₂,…,a_(n) of at least two real numbers is the sum ∑_(i=1)^(n−1)|a_(i+1)−a_(i)|. Suppose x₁,x₂,…,x₉ are chosen uniformly and randomly from the interval [0,1]. Let M be the largest possible value of the spikiness of a permutation of x₁,x₂,…,x₉. Compute the expected value of M.
Original LaTeX notation
The spikiness of a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of at least two real numbers is the sum $\sum_{i=1}^{n-1}\left|a_{i+1}-a_{i}\right|$. Suppose $x_{1}, x_{2}, \ldots, x_{9}$ are chosen uniformly and randomly from the interval $[0,1]$. Let $M$ be the largest possible value of the spikiness of a permutation of $x_{1}, x_{2}, \ldots, x_{9}$. Compute the expected value of $M$.

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Source and history

task_id
d91c5523-d848-528c-9c1e-117a2aa5e33e
task_key
test--d91c5523-d848-528c-9c1e-117a2aa5e33e
task_revision_id
1
config / split
/ test

Official source

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