Part CCSIR NET December 2024the-rescaled-beta-becomes-a-gamma

The rescaled beta becomes a gamma

Let $X_{1}, X_{2}, \dots, X_{n} (n \geq 3)$ be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). For r = 2, 3, define Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as $n \to \infty, r = 2, 3$ . Then, which of the following statements are true?

A. $Y_{2}$ follows gamma distribution with $E (Y_{2}) = 2$ .✓
B.E(Y(3,n)) → 3 as $n \to \infty$ ✓
C. $Y_{3}$ follows beta distribution with $E (Y_{3}) = 1/3$ .
D.Y(2,n) follows beta distribution with parameters 2 and n − 1.

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.

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50 are analysed free — try those first.

The rescaled beta becomes a gamma

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