[R] A stupid question on F-statistics

[Feng Zhang]

Dear all,
Just a stupid question confused me for a long time.

Now suppose p_dim random vector x (column vector) are from
a multivariate normal N(mu, Sigma).

Given a sample size n, (x1, x2, ....., xn),
and the sample mean is x_bar, sample covariance is S.
I can infer that
n(x_bar - mu)'*inverse(S)*(x_bar - mu) is distributed as F(p, n-p), and for
a new observation vector x,
(x - mu)'*inverse(S)*(x - mu) is distributed as F(p, n-p).

So why the follow statistics T is distributed as
(n+1)(n-1)p/n(n-1)F(p, n-1)?
T = (x - x_bar)'*inverse(S)*(x - x_bar)

Thanks so much for pointing me the way to solve it out.

Fred