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 $0 < x < y < \infty$ , and 0 otherwise. For n = 1, 2, 3, …, let $N_{n}$ 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?

A. $N_{n} / (3 n)$ converges to e⁻ $^{2}$ with probability one.✓
B. $N_{n}$ converges to e⁻ $^{2}$ in probability.
C. $N_{n} / n$ converges to 3e⁻ $^{2}$ in distribution.✓
D. $(N_{n} - 3$ ne⁻ $^{2}) / 3 n$ converges in distribution to a normal random variable with mean zero and variance e⁻ $^{2} (1 - 3 e$ ⁻ $^{2})$ .✓

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.

The count grows only its proportion converges

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