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Part CCSIR NET December 2023min-of-exponentials

Min of exponentials

Let Xi(i=1,2)X_{i} (i = 1, 2) be independent with densities fi(x)=if_{i}(x) = i e−ixe^{-ix} for x ≥ 0. Consider a series system of two independent components with lifetimes X1X_{1} and X2X_{2}, and let X denote the lifetime of the system. Which of the following statements are true?

  1. A.P(X > 4) = P(X > 1)P(X > 2)
  2. B.P(X > 4 | X > 2) = P(X > 2)✓ Correct
  3. C.E(X) = 1/3✓ Correct
  4. D.6X ~ χ32\chi^{2}_{3}

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: If X and Y are each normal and uncorrelated then they are independent

More on this topic

The chapter behind this: Joint distributions, transformations and order statistics — free to read

From Probability › Joint distributions, transformations, order statistics

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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