  We use the holonomic gradient method to evaluate the probability content of a
simplex region under a multivariate normal distribution. This probability
equals to the integral of the probability density function of the multivariate
Gaussian distribution on the simplex region. For this purpose, we generalize
the inclusion--exclusion identity which was given for polyhedra, to the faces
of a polyhedron. This extended inclusion--exclusion identity enables us to
calculate the derivatives of the function associated with the probability
content of a polyhedron in general position. We show that these derivatives can
be written as integrals of the faces of the polyhedron.
