Let $X_1, X_2, \ldots, X_n$ be independent random variables such that
$$X_i \sim \mathcal{N}(0,1)$$
Define the sample mean as
\[
\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i.
\]
Which of the following statement(s) is/are correct?
- $\displaystyle \sum_{i=1}^{n} X_i^2 \sim \chi^2(n)$
- $(\sqrt{n}\,\bar{X})^2 \sim \chi^2(2)$
- $X_1^2 + X_2^2$ follows an exponential distribution with mean $2$
- $\displaystyle \sum_{i=1}^{n} (X_i - \bar{X})^2 \sim \chi^2(n-1)$
