Math Dopers: What's the distribution of sqrt(X^2+Y^2) if X&Y are Gaussian and correlated?

Factual Questions
1

if Z = sqrt(X^2+Y^2) , and X and Y are zero-mean, iid, normally-distributed random variables, then Z is Rayleigh distributed.

What is the distribution of Z if X and Y are correlated (but still zero-mean, normally-distributed random variables)?

Can’t seem to find info on this scenario, even though it should be relatively common

2

How can they be random and correlated?

3

There are plenty of ways they can be random and correlated. In fact, that’s the whole problem: We don’t know how they’re correlated, so we can’t answer the question.

4

I assume the OP wants X and Y to be bivariate normals so the correlation wold be described simply by the single parameter.

But even given that, I don’t know the answer.

5

The quantity X^2 + Y^2 follows a generalized chi-squared distribution. I don’t think there’s a common name for the distribution of the square root.

6

Presumably, but we don’t know the parameter.

7

I was wondering whether there was a closed-form expression for a general correlation of rho.

It’s looking a little unlikely at this point

8

There’s a closed form for the pdf in terms of the Bessel function BesselI(0,r) [that’s a capital-i, not a small-L], for what it’s worth: Diagonalize the covariance and rescale so that you have uncorrelated Gaussian random variables U=N(0,1) and V=N(0,s[sup]2[/sup]) with s<=1, and Z=sqrt(U[sup]2[/sup]+V[sup]2[/sup]). (The smaller of the two principal-axis variances s[sup]2[/sup] will be our extra parameter.)

Converting to polar coordinates and doing the angular integral gives the pdf,
f(z) = (z/s) Exp[-(z[sup]2[/sup]/4)(1/s[sup]2[/sup]+1)] BesselI(0,(z[sup]2[/sup]/4)(1/s[sup]2[/sup]-1)) .

Note that when s=1 you have the isotropic case and the distribution becomes Rayleigh; when s<<1 the distribution converges to the right half of the normal distribution.