Student's Theorem

Definition

Let $X_i\sim N(\mu ,{\sigma }^2)$ be i.i.d. Random Vector following Normal Distribution, $\overline{X}:=\frac{1}{n}\sum _{i=1}^n{X}_i$, and $s^2:=\frac{1}{n-1}\sum _{i=1}^n(X_i-\overline{X}{)}^2$, then

• $\overline{X}\sim N(\mu ,\frac{{\sigma }^2}{n})$
• $\overline{X}$ and $s^2$ are independent
• $\frac{1}{{\sigma }^2}\sum _{i=1}^n(X_i-\overline{X}{)}^2=\frac{(n-1)s^2}{{\sigma }^2}\sim {\chi }^2(n-1)$
• $\frac{\overline{X}-\mu }{s/\sqrt{n}}\sim t(n-1)$