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

Basu makes the ratio independent

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

A. $C o v (X_{(n)} / X_{(1)}, X_{(n)}) = 0$ ✓
B. $E (X_{(1)} / X_{(n)}) = E (X_{(1)}) / E (X_{(n)})$ ✓
C. $C o v (X_{(1)} / X_{(n)}, X_{(n)}) = 0$ ✓
D. $C o v (ln (X_{(1)}) - ln (X_{(1)} + X_{(n)}), X_{(n)}) < 0$

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.

See pricing

50 are analysed free — try those first.

Related counterexample: A sufficient statistic is complete

Basu makes the ratio independent

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