Variance of a vector product/sum combination

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Discussion Overview

The discussion revolves around finding the variance of the expression y*(s+n), where y is a vector following a chi-squared distribution, and s and n are vectors following Gaussian distributions. The scope includes theoretical aspects of variance in the context of random vectors and their distributions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks assistance in calculating the variance of the expression involving random vectors y, s, and n.
  • Another participant questions the dimensionality of the vectors, noting that the product y*(s+n) may not be defined if y is a m x 1 vector.
  • A participant corrects the dimensionality, clarifying that y should be a 1 x m vector instead.
  • There is a query regarding the independence of the vectors and the nature of their variance/covariance matrices.
  • A participant confirms the independence of the vectors and specifies the variance/covariance matrices of s and n.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the method to calculate the variance, and multiple viewpoints regarding the definitions and properties of the vectors remain present.

Contextual Notes

There are unresolved questions regarding the assumptions of independence and the specific forms of the variance/covariance matrices for the vectors involved.

nikozm
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Hi,

i am trying to find the variance of the following: y*(s+n), where y is a m \times 1 vector following a chi-squared distribution with 2k degrees of freedom, s is a m \times 1 vector following a Gaussian distribution with zero mean and unit-variance, and n is a m \times 1 vector following a Gaussian distribution with zero mean and variance z.

Any help would be useful
 
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If y is [itex]m \times 1[/itex] and both s, n are also [itex]m \times 1[/itex], the product [itex]y \cdot (s + n)[/itex] is not defined: what are you trying to do?
 
Sorry for this typo. Let y be a 1 \times m vector instead.

Thanks
 
Are they independent? and do you mean the variance/covariance matrix of [itex]s[/itex] is the identity matrix and that of [itex]n[/itex] is [itex]z[/itex] times the identity matrix?
 
yes. and they are mutually independent random vectors. Do you have any clue ?
 

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