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Omni-MATH / Suppose that X₁,X₂,… are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑_(i=1)^(k)X_(i)/2^(i), where k is the least positive integer suc…

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problem

Suppose that X1,X2,…X_1, X_2, \dots are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑i=1kXi/2iS = \sum_{i=1}^k X_i/2^i, where kk is the least positive integer such that Xk<Xk+1X_k < X_{k+1}, or k=∞k = \infty if there is no such integer. Find the expected value of SS.
Plain-text mathematical notation (without MathML)
Suppose that X₁,X₂,… are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑_(i=1)^(k)X_(i)/2^(i), where k is the least positive integer such that X_(k)<X_(k+1), or k=∞ if there is no such integer. Find the expected value of S.
Original LaTeX notation
Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$.

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Source and history

task_id
b45274e7-ad8d-56bd-90f4-b7b72c66b201
task_key
test--b45274e7-ad8d-56bd-90f4-b7b72c66b201
task_revision_id
1
config / split
/ test

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