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!

 

[blue_raver22]

Hello Alex,

Thank you for your question regarding the product of a Gaussian random variable with itself. This is a very interesting topic and one that has been studied extensively in the field of statistics.

Firstly, to answer your question, the product of a Gaussian random variable with itself is not necessarily a chi-square distribution. This only holds true if the two Gaussian variables are independent. In the case of multiplying a variable with itself, it is not independent and therefore the resulting distribution will not be a chi-square.

To better understand the distribution of X^2, we can look at the properties of a Gaussian random variable. A Gaussian random variable, also known as a normal distribution, is characterized by its mean and standard deviation. When we square a Gaussian variable, we are essentially multiplying the mean and standard deviation by themselves. This results in a new distribution with a higher variance and a non-zero mean.

The resulting distribution is known as a non-central chi-square distribution, which is a special case of the chi-square distribution. It is used to model the sum of squared independent Gaussian variables with non-zero means. The degree of freedom for this distribution is equal to the number of squared Gaussian variables being added together.

In summary, the product of a Gaussian random variable with itself is not a chi-square distribution, but rather a non-central chi-square distribution. It is important to note that this only holds true for Gaussian variables. If the variables are not Gaussian, the resulting distribution may be different.

I hope this helps clarify your question. If you have any further questions or concerns, please do not hesitate to reach out.

Best,