Let $X_{1}, X_{2}, \dots, X_{n}$ be random sample from Normal population with mean $\mu$ and variance $\sigma^{2}$. Then which of the following results are correct?
A. $\overline{X}\sim N(\mu,\frac{\sigma^{2}}{n})$
B. $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}\sim\chi_{n}^{2}$
C. $\overline{X}$ and $\sum_{i=1}^{n}(\frac{X_{i}-\overline{X}}{\sigma})^{2}$ are independently distributed
D. $\frac{(\overline{X}-\mu)^{2}}{\frac{\sigma^{2}}{n}}\sim \chi_{1}^{2}$
E. $\sum_{i=1}^{n}(\frac{X_{i}-\mu}{\sigma})^{2}\sim \chi_{n-1}^{2}$
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Remember: Using the sample mean $\overline{X}$ instead of the population mean $\mu$ in the Chi-square sum "costs" you exactly one degree of freedom. This is why the sum involving $\overline{X}$ has $n-1$ degrees of freedom.