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

The rescaled beta becomes a gamma

Let X1,X2,…,Xn(n≥3)X_{1}, X_{2}, \dots, X_{n} (n \ge 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→∞,r=2,3n \to \infty, r = 2, 3. Then, which of the following statements are true?

  1. A.Y2Y_{2} follows gamma distribution with E(Y2)=2E(Y_{2}) = 2.✓ Correct
  2. B.E(Y(3,n)) → 3 as n→∞n \to \infty✓ Correct
  3. C.Y3Y_{3} follows beta distribution with E(Y3)=1/3E(Y_{3}) = 1/3.
  4. 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, and why this one tests limit assumed to exist.

See pricing

50 are analysed free — try those first.

The trap it tests

Limit assumed to exist

Reasoning with a limit before establishing there is one. limsup is not lim.

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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