Part CCSIR NET June 2025the-variance-of-x-squared-is-the-fourth-moment-minus-1-not-the-fourth-moment

The variance of x squared is the fourth moment minus 1 not the fourth moment

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables with $E (X_{1}) = 0, E (X_{1}^{2}) = 1, E (X_{1}^{3}) = 0, E (X_{1}^{4}) = 3$ . Let $S_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}, T_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{2}, U_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{3}$ and $V_{n} = \sum_{i}$ ₌ $_{1}^{n} X_{i}^{4}$ . Then, which of the following statements are true?

A. $S_{n} / n$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓
B. $(T_{n} - n) / 3 n$ converges in distribution to a random variable Z, where Z ~ N(0, 1)
C. $n S_{n} / T_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 1)✓
D. $(T_{n} - n) / V_{n}$ converges in distribution to a random variable Z, where Z ~ N(0, 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 moments and tails.

See pricing

50 are analysed free — try those first.

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

The variance of x squared is the fourth moment minus 1 not the fourth moment

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