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) = Γ( 1 2 (ν + r))

(πν)r/2Γ(ν/2)

1 + 1

ν

v2 i

− 1 2 (ν+r)

.