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Part CCSIR NET June 2024basu-makes-the-ratio-independent

Basu makes the ratio independent

Let X1,…,XnX_{1}, \dots, X_{n} be independent and identically distributed U(0,θ),θ>0U(0, \theta), \theta > 0 random variables. Define X(n)=max⁡X_{(n)} = \max{X1,…,XnX_{1}, \dots, X_{n}} and X(1)=min⁡X_{(1)} = \min{X1,…,XnX_{1}, \dots, X_{n}}. Which of the following statements are true?

  1. A.Cov(X(n)/X(1),X(n))=0Cov(X_{(n)}/X_{(1)}, X_{(n)}) = 0✓ Correct
  2. B.E(X(1)/X(n))=E(X(1))/E(X(n))E(X_{(1)}/X_{(n)}) = E(X_{(1)})/E(X_{(n)})✓ Correct
  3. C.Cov(X(1)/X(n),X(n))=0Cov(X_{(1)}/X_{(n)}, X_{(n)}) = 0✓ Correct
  4. D.Cov(ln⁡(X(1))−ln⁡(X(1)+X(n)),X(n))<0Cov(\ln(X_{(1)}) - \ln(X_{(1)} + X_{(n)}), X_{(n)}) < 0

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: A sufficient statistic is complete

More on this topic

The chapter behind this: Sufficiency, completeness and the UMVUE machine — free to read

From Estimation › Sufficiency, completeness, UMVUE, Cramér–Rao

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

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