Hi! Can someone explain this to me:

The definition of $\displaystyle \chi^2$-distribution, taken from my statistics book, is:

If $\displaystyle X_1,\ X_2,... ,\ X_f$ are independent and $\displaystyle ~N(0,\ 1)$, then

$\displaystyle \sum_{i=1}^f X_i^2\ \sim\chi^2(f)$

f is the number of degrees of freedom.

The book also says (but it doesn't prove it) that if $\displaystyle X_1,\ X_2,... ,\ X_n$ are independent and $\displaystyle ~N(0,\ 1)$, then

$\displaystyle \sum_{i=1}^n (X_i-\bar{X})^2\ \sim\chi^2(n-1),$ where $\displaystyle \bar{X}=\frac{X_1+X_2+...+X_n}{n}$

I would really like to see what the proof looks like. How can this be proven?