Non central chi square distribution

[bob j]

I read this article about non central chi square distribution
http://en.wikipedia.org/wiki/Noncentral_chi-square_distribution

in practice, if I have a sum of X_i^2, where X_i is gaussian with mean \mu_i and std \sigma_i what would be the pdf of the sum? In the article the assume you have (x_i\sigma_i)^2

 

[Stephen Tashi]

Suppose we have the problem:

Given f( X/2, Y/2) = X + XY
Find f(x,y)

How would you solve it?

You would use the substitution X = 2x, Y = 2y

This is similar to the question that you are asking since the article gives you the pdf for the Xi/sigma_i and you want to find the value of the pdf at the x_i without the divisions by the sigma_i.

Or did you already realize this and were asking someone to do the algebra?

 

[bob j]

why did they bother deriving the generalized chi square distribution then?

 

[Stephen Tashi]

why did they bother deriving the generalized chi square distribution then?

Given that you are going to state a distribution for the sum of squares of independent random variables, it is most natural to state it for variables that have a convenient scale. If you look at data of the form Z_i = (X_i - mu_i)/ sigma_i, you can tell that +4.0 is a very unlikely value and "bigger than average". If look at data that says X_i = 240.0, you have no idea whether this is unusual or whether it is bigger or smaller than average.

 

[bob j]

the gen. chi square looks quite different than the chi square distribution. It's not just scaling, at least from what i can understand

 

[Stephen Tashi]

My remarks relate to scaling the non-central chi-square to find the distribution of a sum of non-normalized non-identically distributed independent normal random variables not to a claim that scaling the chi-square will produce the non-central chi-square. The variables in the chi-square are assumed to be identically distributed.