Question

Let $X_{i} \sim N(0,1), i=1,2, \dots$ be independent random variables. If $T_{n}=\sum_{i=1}^{n} X_{i}^{2}$, then $\lim_{n \rightarrow \infty} P(T_{n} > n+2 \sqrt{2 n})$ is equal to

• A. $\Phi(2)$
• B. $\Phi(-2)$
• C. $1/2$
• D. $1-\Phi(-2)$

Hint:

For large $n$, $\chi^2_n$ is essentially normal. The term $n + k\sqrt{2n}$ represents the mean plus $k$ standard deviations. The probability of being above that is $1-\Phi(k)$.

Answer 1

The Correct Option is B