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MMLU-Pro / 8431 / 5.2-13. Let X₁,X₂ be independent random variables representing lifetimes…

Problem

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

question

5.2-13. Let X1,X2X_1, X_2 be independent random variables representing lifetimes (in hours) of two key components of a device that fails when and only when both components fail. Say each XiX_i has an exponential distribution with mean 1000. Let 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), so that the space of Y1,Y2Y_1, Y_2 is 0<y1<y2<∞ 0< y_1 < y_2 < \infty (a) Find G(y1,y2)=P(Y1≤y1,Y2≤y2)G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right).
Plain-text mathematical notation (without MathML)
5.2-13. Let X₁,X₂ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each X_(i) has an exponential distribution with mean 1000. Let Y₁=min(X₁,X₂) and Y₂=max(X₁,X₂), so that the space of Y₁,Y₂ is 0<y₁<y₂<∞
(a) Find G(y₁,y₂)=P(Y₁≤y₁,Y₂≤y₂).
Original LaTeX notation
5.2-13. Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $
(a) Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$.

category

math

options

  1. 0.6532
  2. 0.7290
  3. 0.2984
  4. 0.6123
  5. 0.8441
  6. 0.4678
  7. 0.5117
  8. 0.3815
  9. 0.4356
  10. 0.5789

src

scibench-stat

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Source and history

upstream_id / Original problem ID
8431
task_id
8df73bb3-d4a9-5b67-9d55-2f80a01e9948
task_key
default--test--8431
task_revision_id
1
config / split
default / test

Official source

initial import