Part CCSIR NET June 2024divide-by-the-right-power-of-n

Divide by the right power of n

Let { $X_{n}$ } $_{n} \geq_{1}$ be a sequence of independent and identically distributed random variables with $E (X_{1}) = 0$ and $V a r (X_{1}) = 1$ . Which of the following statements are true?

A. $lim (n \to \infty) P ((n \cdot \sum_{i}$ ₌ $_{1}^{n} X_{i}) / (\sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}) \leq 0) = 1/2$ ✓
B. $(\sum_{i}$ ₌ $_{1}^{n} X_{i}) / (\sum_{i}$ ₌ $_{1}^{n} X_{i}^{2})$ converges in probability to 0 as $n \to \infty$ ✓
C. $(1/ n) \sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}$ converges in probability to 1 as $n \to \infty$ ✓
D. $lim (n \to \infty) P ((\sum_{i}$ ₌ $_{1}^{n} X_{i}) / n \leq 0) = 1/2$ ✓

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.

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50 are analysed free — try those first.

Divide by the right power of n

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