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Part CCSIR NET December 2024the-count-grows-only-its-proportion-converges

The count grows only its proportion converges

Let {(X_k, Y_k)}(k≥1) be a sequence of independent and identically distributed (i.i.d.) random vectors with common joint probability density function f(x, y) = e^(−y) if 0<x<y<∞0 < x < y < \infty, and 0 otherwise. For n = 1, 2, 3, …, let NnN_{n} be a random variable denoting the number of elements in the set {k : k = 1, 2, …, n; Y_k ≥ 2}. Then, which of the following statements are true?

  1. A.Nn/(3n)N_{n}/(3n) converges to e⁻2^{2} with probability one.✓ Correct
  2. B.NnN_{n} converges to e⁻2^{2} in probability.
  3. C.Nn/nN_{n}/n converges to 3e⁻2^{2} in distribution.✓ Correct
  4. D.(Nn−3(N_{n} - 3ne⁻2)/3n^{2})/\sqrt{3n} converges in distribution to a normal random variable with mean zero and variance e⁻2(1−3e^{2}(1 - 3e⁻2)^{2}).✓ 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 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: Convergence in probability implies almost sure convergence

More on this topic

The chapter behind this: Modes of convergence and the limit theorems — free to read

From Limit Theorems and Markov Chains › Modes of convergence, WLLN, SLLN, CLT

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

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