Let X1,..., Xr be i.i.d. N(0, 1), and let S2 be independent of the X’s and distributed as χ2

ν . Then the distribution of (X1/S

√ν,..., Xr /S

√ν) is a central multivariate t-distribution, and its density is p(v1,...,vr) = ( 1 2 (ν + r))

(πν)r/2(ν/2)



1 +

1

ν

v2 i

− 1 2 (ν+r)

.