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Omni-MATH / A sequence y₁,y₂,…,y_(k) of real numbers is called …k=1, or if y₂−y₁,y₃−y₂,…,y_(k)−y_(k−1) are nonzero and alternate in sign. Let X₁,X₂,…,X_(n) be chosen independently from the uni…

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problem

A sequence y1,y2,…,yky_1,y_2,\dots,y_k of real numbers is called \emph{zigzag} if k=1k=1, or if y2−y1,y3−y2,…,yk−yk−1y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1} are nonzero and alternate in sign. Let X1,X2,…,XnX_1,X_2,\dots,X_n be chosen independently from the uniform distribution on [0,1][0,1]. Let a(X1,X2,…,Xn)a(X_1,X_2,\dots,X_n) be the largest value of kk for which there exists an increasing sequence of integers i1,i2,…,iki_1,i_2,\dots,i_k such that Xi1,Xi2,…,XikX_{i_1},X_{i_2},\dots,X_{i_k} is zigzag. Find the expected value of a(X1,X2,…,Xn)a(X_1,X_2,\dots,X_n) for n≥2n \geq 2.
Plain-text mathematical notation (without MathML)
A sequence y₁,y₂,…,y_(k) of real numbers is called \emph{zigzag} if k=1, or if y₂−y₁,y₃−y₂,…,y_(k)−y_(k−1) are nonzero and alternate in sign. Let X₁,X₂,…,X_(n) be chosen independently from the uniform distribution on [0,1]. Let a(X₁,X₂,…,X_(n)) be the largest value of k for which there exists an increasing sequence of integers i₁,i₂,…,i_(k) such that X_(i₁),X_(i₂),…,X_(i_(k)) is zigzag. Find the expected value of a(X₁,X₂,…,X_(n)) for n≥2.
Original LaTeX notation
A sequence $y_1,y_2,\dots,y_k$ of real numbers is called \emph{zigzag} if $k=1$, or if $y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\dots,i_k$ such that $X_{i_1},X_{i_2},\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\dots,X_n)$ for $n \geq 2$.

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task_id
ff4b5898-b392-553f-a0d3-8c4b7529f7b2
task_key
test--ff4b5898-b392-553f-a0d3-8c4b7529f7b2
task_revision_id
1
config / split
/ test

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