Product of gaussian random variable with itself

[architect]

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

 

[statdad]

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).

 

[Redbelly98]

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.

 

[architect]

thanks for your replies. I will give it a try!