benchmarks.wiki

benchmarks.wiki / Public workspace

MMLU-Pro / 12119 / A random variable R has a Rayleigh density f(r) = (r / …

Problem

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

question

A random variable R has a Rayleigh density f(r) = (r / \alpha^2)e[{-(r)2} / {2(\alpha)2}],0 \leq r \leq \infty and = 0,r < 0 Determine the following: (a) the expected value or R, E [R] (b) the expected value of R^2, E [R^2] and (c) the variance \sigma^2 of R

category

engineering

options

  1. E[R] = \sqrt{\pi}\alpha/2, E[R^2] = \pi\alpha^2/2, \sigma_r^2 = (\pi - 3)\alpha^2/2
  2. E[R] = 2lpha^2, E[R^2] = [\surd(\pi / 2)] ullet lpha, \sigma_r^2 = lpha^2(0.43)
  3. E[R] = \alpha(1 - e^{-1/2}), E[R^2] = \alpha^2, \sigma_r^2 = \alpha^2(e - 2)
  4. E[R] = lpha^2(0.43), E[R^2] = [\surd(\pi / 2)] ullet lpha, \sigma_r^2 = 2lpha^2
  5. E[R] = \alpha\sqrt{e}, E[R^2] = \alpha^2\ln(2), \sigma_r^2 = \alpha^2(2 - e/\pi)
  6. E[R] = [\surd(\pi / 2)] ullet lpha, E[R^2] = 2lpha^2, \sigma_r^2 = lpha^2(0.43)
  7. E[R] = [\surd(\pi / 2)] ullet lpha, E[R^2] = lpha^2(0.43), \sigma_r^2 = 2lpha^2
  8. E[R] = \alpha^2, E[R^2] = (2/3)\alpha^2, \sigma_r^2 = (4 - \pi)\alpha^2/2
  9. E[R] = \alpha/\sqrt{2}, E[R^2] = \alpha^2, \sigma_r^2 = \alpha^2(\pi - 2)/2
  10. E[R] = \alpha\sqrt{2/\pi}, E[R^2] = 3\alpha^2/2, \sigma_r^2 = \alpha^2(1 - 2/\pi)

src

stemez-ElectronicCommunications

Discussion

Discussion

No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

Source and history

upstream_id / Original problem ID
12119
task_id
d3a2e6de-40c2-5e9c-9f40-8878995d6bce
task_key
default--test--12119
task_revision_id
1
config / split
default / test

Official source

initial import