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

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The discussion centers on the distribution of Z, defined as Z=sqrt(X^2 + Y^2), where X and Y are normally distributed with different means and variances. For Z to follow specific distributions like Chi, Rayleigh, or Ricean, certain conditions regarding the means and variances must be met. In the case presented, both means and variances differ, complicating the determination of Z's distribution. It is suggested that Z may follow a non-central chi distribution, where means can vary, but there is uncertainty regarding the variances. The conversation highlights the complexity of deriving Z in error analysis and points to numerical methods for handling distributions of non-central chi-squared variables.
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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.
 
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Plz help!
 
It's a non-central chi distribution.
 
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.
 
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.
 
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)
 
thank you so much. I'll look it up.
 
How is the Z derived in the error analysis?
 

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