What is this distribution?

In summary: However, the result won't be the same as the distribution of the square of a standard normal random variable.In summary, the conversation starts with a question about the distribution of a square of a sum of two Gaussian variables, with a link to an attached image. The poster then asks for the name and properties of this distribution. Another poster suggests that the distribution could be a chi square distribution with one degree of freedom. However, the original poster clarifies that the variables may have different variances. The conversation then moves to discussing the derivation of the distribution and whether or not it has a nice name or form. Finally, the original poster mentions a related discussion and presents a different question about the distribution of a Gaussian squared.
  • #1
nem0
4
0
Hello everyone (first post here :)),

I have a question regarding Probability. I calculated the distribution of a square of a sum of two Gaussian variables and I have got this expression (I attached a link to it). (Gaus1+Gaus2)²
It is like a normal Gaussian distribution (of x) only divided by the square root of the argument x.
http://imageshack.us/photo/my-images/171/distribg.png/

My question is: Does this distribution have a name? I want to check its properties on wiki but I don't know its name.

Thanks a lot!

P.S. (if the image linking does not work just copy this link into your browser)
 
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  • #2
What you calculated doesn't look like a probability distribution. Are you saying that the integral of that function with respect to r, from zero to infinity is 1?

The sum of two independent gaussian random variables is another gaussian random variable. So I think you are asking for the distribution for the square of a single gaussian random variable, which is a chi square distribution with one degree of freedom.
 
  • #3
Well I think the variable r should go from 0 to infinity and it does sum up to 1 (since r is actually X² so it should not be negative).
I have used the formula that is to be found on
http://www.math.dartmouth.edu/~prob/prob/prob.pdf

page 303/517 (I'm referring to pages of pdf document, not the pages in book)
there is an expression for calculation of square of a RV. I plugged in the expression for X which is the sum of 2 Gaussians et voilà.

I think it is fine since it is in accordance with some results I found in some scientific papers, but then again I might be wrong. Probability is not actually my field :)
 
  • #4
(Gaus1+Gaus2)²

nem0 said:
I have used the formula that is to be found on
http://www.math.dartmouth.edu/~prob/prob/prob.pdf
page 303/517

If X and Y are the two unit normal random variables, you say in the original post that your want the distribution of [itex] (X + Y)^2 [/itex] but on page 303 of that document you seem to have used the formula for the Rayleigh distribution, which is the distribution for [itex] (X^2 + Y^2) ^ {\frac{1}{2}} [/itex].
 
  • #5
Actually when you say that the square of Gaussians gives a chi square it is true if the Gaussians are standard normal (zero mean variance 1), but if they have variance different than 1 then I don't know if this thing holds. Anyway if you take the thing that I derived and put variance equal to one I think you will get Chi square.

Maybe I should rephrase my Q:
What is the distribution (of a sum) of squares of non-standard (variance not equal 1) Gaussian variables?
Or maybe the answer I already have here:
Say I have Gaussian RV N~(0, sigma).
If I want the distribution of Gaussian²:
find the distribution of Gaussian²/sigma (it will be Chi², since it's standard Gaussian - its been normalized)
its statistic properties are (mean, variance)
so statistic properties of the wanted (Gaussian²) are (mean*sigma, variance*sigma²)

And about the derivation itself, starting expression is actually to be found under Theorem 5.1 (page 219/518). Sorry.
 
  • #6
wow its nice u can insert equations here! :)
 
  • #7
You aren't being clear. Are your asking several different questions?

Your original post asks about summing gaussian variables and then squaring the sum. I'll assume we aren't dealing with that anymore.

nem0 said:
What is the distribution (of a sum) of squares of non-standard (variance not equal 1) Gaussian variables?

That's one question. I don't know if such a distribution has a nice name or form.
A related discussion is: https://www.physicsforums.com/showthread.php?t=507750

Or maybe the answer I already have here:
Say I have Gaussian RV N~(0, sigma).
If I want the distribution of Gaussian²:
find the distribution of Gaussian²/sigma (it will be Chi², since it's standard Gaussian - its been normalized)

That's a different question and I agree you can use a change of variable to transform the distribution of the square of a non-mean-zero non-unit-variance normal random variable to a chi square distribution.
 

What is this distribution?

This is a common question asked by scientists when analyzing data. A distribution is a way of showing how often certain values occur in a set of data. It is essentially a representation of the frequency or probability of different outcomes in a data set.

What are the types of distributions?

There are several types of distributions, including normal, binomial, Poisson, and exponential. Each type of distribution has its own characteristics and is used to model different types of data.

How do I determine the shape of a distribution?

The shape of a distribution can be determined by looking at the data visually, using a graph or histogram, or by calculating measures of central tendency and variability such as mean, median, and standard deviation. These measures can give an indication of whether the distribution is symmetrical, skewed, or has multiple peaks.

What is the purpose of understanding distributions?

Understanding distributions is important in statistics and data analysis as it allows us to make conclusions about a population based on a sample. Distributions can also help us identify outliers and make predictions about future data.

How can I use distributions in my research?

Distributions can be used in various ways in scientific research, such as analyzing the results of experiments, comparing groups or populations, and making predictions about future data. They can also be used to test hypotheses and make inferences about a larger population.

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