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MMLU-Pro / 7782 / 5.3-19. Two components operate in parallel in a device, so the device fails when…

Problem

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

question

5.3-19. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, X1X_1 and X2X_2, of the respective components are independent and identically distributed with an exponential distribution with θ=2\theta=2. The cost of operating the device is Z=2Y1+Y2Z=2 Y_1+Y_2, where Y1=min(X1,X2)Y_1=\min \left(X_1, X_2\right) and Y2=max(X1,X2)Y_2=\max \left(X_1, X_2\right). Compute E(Z)E(Z).
Plain-text mathematical notation (without MathML)
5.3-19. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, X₁ and X₂, of the respective components are independent and identically distributed with an exponential distribution with θ=2. The cost of operating the device is Z=2Y₁+Y₂, where Y₁=min(X₁,X₂) and Y₂=max(X₁,X₂). Compute E(Z).
Original LaTeX notation
5.3-19. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$.

category

math

options

  1. $9$
  2. $8$
  3. $4$
  4. $5$
  5. $12$
  6. $3$
  7. $10$
  8. $4.5$
  9. $7$
  10. $6$

src

scibench-stat

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Source and history

upstream_id / Original problem ID
7782
task_id
6a901482-603d-5a52-80cb-5c447162cfce
task_key
default--test--7782
task_revision_id
1
config / split
default / test

Official source

initial import