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

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    Chi Rayleigh
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Discussion Overview

The discussion revolves around the distribution of the random variable Z, defined as Z=sqrt(X^2 + Y^2), where X and Y are normally distributed with different means and variances. Participants explore the implications of differing parameters on the distribution of Z, particularly in the context of error analysis.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that for Z to follow a Chi distribution, both means must be zero and variances must be one; for a Rayleigh distribution, means must be zero and variances must be equal; for a Ricean distribution, means can differ but variances must be equal.
  • Another participant suggests that Z follows a non-central chi distribution, indicating that means can differ.
  • A subsequent reply questions whether variances can also differ in a non-central chi distribution, stating that variances must be one based on their understanding.
  • One participant acknowledges the complexity of the situation, suggesting that a nice analytic distribution for Z may not exist and mentions numerical methods for handling distributions of linear combinations of non-central chi squared random variables.
  • Another participant expresses gratitude for the information and indicates a willingness to research further.
  • A later post inquires about the derivation of Z in the context of error analysis.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which Z can be classified under various distributions, particularly regarding the means and variances. The discussion remains unresolved regarding the exact distribution of Z when both means and variances differ.

Contextual Notes

There are limitations in the discussion regarding assumptions about the distributions and the specific conditions under which Z can be classified. The mathematical steps leading to the characterization of Z are not fully resolved.

ay0034
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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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