Mathematics · Probability Theory
Let {(Xk, Yk)}k=1∞ be a sequence of independent and identically distributed (i.i.d.) random vectors with joint probability density function f(x, y) = e-y for 0 < x < y < ∞ and 0 otherwise. For n = 1, 2, ..., let Nn be the number of elements in {k : 1 ≤ k ≤ n, Yk ≥ 2}. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1Nnn converges to 2e-2 almost surely (with probability 1).✓
- 2Nn converges to e-2 in probability.
- 3Nnn converges to 3e-2 in distribution.
- 4Nn - 2ne-2√(n) converges in distribution to a normal variable.
Correct answer
Nnn converges to 2e-2 almost surely (with probability 1).
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