Question

Let $X_{1}, X_{2}, \dots$ be independent variables each taking values $+1$ or $-1$ with equal probability respectively. If $S_{n}=\sum_{i=1}^{n} i X_{i}$ then $\lim_{n \rightarrow \infty} P\left(S_{n} < \sqrt{\frac{n(n+1)(2 n+1)}{3}}\right)$ where $\Phi$ is distribution function of standard normal variate, is

• A. $\Phi(-\sqrt{3})$
• B. $\Phi(-\sqrt{2})$
• C. $1-\Phi(-\sqrt{3})$
• D. $1-\Phi(-\sqrt{2})$

Hint:

For any sum $S_n$, the limit of $P(S_n < k \sigma_n)$ is always $\Phi(k)$. Identify the variance $\sigma_n^2$ first to see how many standard deviations the bound represents.

Answer 1

The Correct Option is C