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MMLU-Pro / 8312 / Let X_2,X_3,... be independent random variables such that…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let X_2,X_3,... be independent random variables such that $P(X_n=n)=P(X_n=-n)=1/(2n\log (n)), P(X_n=0)=1-1/(n*\log(n))$. Does n−1∑i=2nXin^{-1}\sum_{i=2}^n X_i converges in probability? Does n−1∑i=2nXin^{-1}\sum_{i=2}^n X_i converges in almost surely? Return the answers of the two questions as a list.
Plain-text mathematical notation (without MathML)
Let X_2,X_3,... be independent random variables such that $P(X_n=n)=P(X_n=-n)=1/(2n\log (n)), P(X_n=0)=1-1/(n*\log(n))$. Does n^(−1)∑_(i=2)^(n)X_(i) converges in probability? Does n^(−1)∑_(i=2)^(n)X_(i) converges in almost surely? Return the answers of the two questions as a list.
Original LaTeX notation
Let X_2,X_3,... be independent random variables such that $P(X_n=n)=P(X_n=-n)=1/(2n\log (n)), P(X_n=0)=1-1/(n*\log(n))$. Does $n^{-1}\sum_{i=2}^n X_i$ converges in probability? Does $n^{-1}\sum_{i=2}^n X_i$ converges in almost surely? Return the answers of the two questions as a list.

category

math

options

  1. [0.5, 0]
  2. [1, 0]
  3. [0, 0.5]
  4. [0.5, 1]
  5. [1, -1]
  6. [0, 1]
  7. [0.5, 0.5]
  8. [1, 1]
  9. [1, 0.5]
  10. [0, 0]

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8312
task_id
d4cbdf95-1b9e-5c4c-84ff-59c9128adba8
task_key
default--test--8312
task_revision_id
1
config / split
default / test

Official source

initial import