Has anybody come across the following representation of a Beta distribution:

If Z1, Z2, ... Zn are independent identically distributed Standard Normal random variables (i.e., Zi~N(0,1) and iid for i=1,2,..,n) Then,

Y=(Z1+Z2+ ... +Zn)^2/[n*(Z1^2+Z2^2+ ... +Zn^2)]

is distributed as Beta(1/2,(n-1)/2). I have done some simulations and this appears to be true. However I cannot find a reference to this characterisation of a Beta distribution in terms of Standard Normal variables. Can you point me to a reference if you have seen this characterisation?

Or have I discovered something new?