What Distribution Does Z Follow When Both Means and Variances Differ?

[ay0034]

Hello all,

I've been working on error analysis of the system, and I finally faced a big problem.

Let X~N(mu1, sigma1^2) and Y~N(mu2, sigma2^2), and Z=sqrt( X^2 + Y^2 )


For Z to be a Chi, mu's should be zero and sigma's should be 1, to be a Rayleigh, mu's should be zero and two sigma's should be the same, and finally to be a Ricean, mu's can be different from each other, but two sigma's should be the same.

Yes, that's all I know. But in my case, mu's are different and sigma's are different as well. In this case, what is 'Z'?

I appreciated it in advance.

 

[ay0034]

Plz help!

 

[sfs01]

It's a non-central chi distribution.

 

[ay0034]

For a non-central chi distribution, means can be different. But can variances be different as well? As far as I know, variances must be 1.

 

[sfs01]

Err, sorry, got a bit over-enthusiastic and thought it should fit nicely, but seems it doesn't after all, so yes looks more complicated.

 

[sfs01]

And don't think you'll get a nice analytic distribution for this. There seem to be a few numerical methods out there for managing the distribution of a linear combination of non-central chi squared random variables though, which is fairly close to what you want (apart from the square root)

 

[ay0034]

thank you so much. I'll look it up.

 

[bpet]

How is the Z derived in the error analysis?