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