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Part CCSIR NET June 2024orderings-are-distribution-free

Orderings are distribution free

Let X1,X2,X3X_{1}, X_{2}, X_{3} be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?

  1. A.P(max{X1,X2X_{1}, X_{2}} < 1) = 1/(2e)
  2. B.P(min{X1,X2X_{1}, X_{2}} > 1) = 1/(2e)
  3. C.P(min{X1,X2X_{1}, X_{2}} <X3)=2/3< X_{3}) = 2/3✓ Correct
  4. D.P(max{X1,X2X_{1}, X_{2}} <X3)=1/3< X_{3}) = 1/3✓ Correct

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 standard counterexample.

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50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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

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