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

## distribution of a quadratic function of a multivariate normal

**Moderators:** gmalivuk, Moderators General, Prelates

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

I found the pictures from this lecture to be helpful in the case of a 1-dimensional random variable: https://www.cis.rit.edu/class/simg713/Lectures/Lecture713-04.pdf. I haven't quite figured out a good way to compute this kind of thing for multivariate functions, but maybe the univariate case can be interesting.

Let's see what happens to a univariate normal distribution for starters, so z=x

Going to the multivariate case, you could do a change-of-variables y=P

Let's see what happens to a univariate normal distribution for starters, so z=x

^{2}with x having a normal distribution with mean μ and variance σ. Working through the math, I get that the pdf of z is f(z)=(8πσ^{2}z)^{-1/2}(exp[-(sqrt(z)-μ)^{2}/(2σ^{2})]+exp[-(-sqrt(z)-μ)^{2}/(2σ^{2})]). Whatever this is called, it looks something like this. You see a spike at 0, and sometimes another hump depending on the μ and σ you pick.Going to the multivariate case, you could do a change-of-variables y=P

^{1/2}x if P is positive semidefinite. This may let you integrate a Gaussian over the level sets of y^{T}y, which is a hypersphere. I'm not sure how doable this is, since this may be some nasty integral in several dimensions. The situation could be worse if P isn't positive semidefinite because then the level sets may be unbounded and I have no idea how to parameterize these kinds of surfaces. It should be simple enough to run a bunch of simulations and draw a histogram, and I suspect the distribution will still have that spike at zero.### Re: distribution of a quadratic function of a multivariate normal

So I took the easy way out and went googling. I was actually pretty skeptical there would be a solution to this, mainly because I have seen derivations for the expectation and covariance of the expression many times, but I have never seen an attempt to figure out what the distribution actually is.

So I was pretty surprised when I found this,

http://www-1.ms.ut.ee/tartu07/presentations/ohlson.pdf (slide 18/19)

which tells us that the result is some sort of chi-squared or sum of chi-squared variables.

So I was pretty surprised when I found this,

http://www-1.ms.ut.ee/tartu07/presentations/ohlson.pdf (slide 18/19)

which tells us that the result is some sort of chi-squared or sum of chi-squared variables.

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