Product of gaussian random variable with itself

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Discussion Overview

The discussion revolves around the properties of the product of a Gaussian random variable with itself, specifically examining what the distribution of X^2 is when X is a Gaussian variable. Participants explore whether this results in a chi-square distribution or another form.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Alex questions the nature of the variable X^2 when X is Gaussian, suggesting uncertainty about whether the result is still Gaussian or if it is a chi-square distribution.
  • Another participant notes that if X is a standard Gaussian, then Y = X^2 will only take non-negative values, implying it cannot be standard Gaussian and suggests investigating chi-square distributions.
  • A separate contribution discusses the conservation of probability and provides a general approach to finding the probability density function of a transformed variable, though it does not directly address the specific case of X^2.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the distribution of X^2, with multiple viewpoints presented regarding its classification and properties.

Contextual Notes

The discussion includes assumptions about the properties of Gaussian variables and their transformations, but these assumptions are not fully explored or resolved.

Who May Find This Useful

Readers interested in probability theory, statistical distributions, or the properties of Gaussian random variables may find this discussion relevant.

architect
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Hi,

I am interested in the product of a Gaussian random variable with itself. If X is Gaussian then what is X^2? We know that the resultant variable of the product of two independent Guassian variables is still Gaussian but I am afraid that this is not true when you multiply it with itself. Is it a chi-square? Any clarifications will be appreciated.

BW,

Alex
 
Last edited:
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architect said:
Hi,

I am interested in the product of a Gaussian random variable with itself. If X is Gaussian then what is X^2? We know that the resultant variable of the product of two independent Guassian variables is still Gaussian but I am afraid that this is not true when you multiply it with itself. Is it a chi-square? Any clarifications will be appreciated.

BW,

Alex

Partial answer to get you thinking more: If, as a specific case, X is standard Gaussian, notice that Y = X^2 will take on only non-negative values so certainly could not be standard Guassian. If I were you I'd investigate the origins of chi-square distributions (both central and non-central chi-square).
 
If p(x) is the probability density function of x, then conservation of probability tells us that

p(x) dx = q(y) dy​

where y is any known function of x. Solve the equation for q(y) to get the probability density function for y.
 
thanks for your replies. I will give it a try!
 

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