### distribution of a quadratic function of a multivariate normal

Posted:

**Sat Jul 01, 2017 7:10 pm UTC**if x is drawn from a multivariate normal with mean mu and covariance sigma, what's the distribution of z = x.T P x

(.T denotes transpose and P is an arbitrary square matrix)

i suspect that f(z) = ∫ p(x) dx

where the integral is over all x such that x.T P x = z

but i'm not sure how to carry out this integral over that strange region

for the special case that P = sigma^-1 = identity, then z is the sum of squares of normals, which is chi-squared

possibly this problem can also be solved for P = sigma^-1 != I, but i'm not sure how

i would also be satisfied with knowing if there is no nice expression for the resulting distribution

(.T denotes transpose and P is an arbitrary square matrix)

i suspect that f(z) = ∫ p(x) dx

where the integral is over all x such that x.T P x = z

but i'm not sure how to carry out this integral over that strange region

for the special case that P = sigma^-1 = identity, then z is the sum of squares of normals, which is chi-squared

possibly this problem can also be solved for P = sigma^-1 != I, but i'm not sure how

i would also be satisfied with knowing if there is no nice expression for the resulting distribution