The discussion revolves around finding the probability density function (PDF) of the sum of "n" independent and identically distributed (iid) non-central chi-square random variables. Participants are exploring the characteristics and relationships of these distributions.
There is an ongoing exploration of different methods to tackle the problem, with participants suggesting various approaches and questioning the relationships between the functions involved. Some guidance has been offered regarding starting points and methods, but no consensus has been reached.
Participants mention that they are learning the basics and express uncertainty about the next steps in their reasoning. There is a reference to a previous thread that may provide additional context or methods.
deema_master said:Homework Statement
I need to find the pdf of sum of "n" iid non central chi-square distribution.
Homework Equations
The PDF of the non-central chi-square RV is given herehttps://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
The Attempt at a Solution
i tried to find the product of the characteristics function of "n" RV, but how to continue?
I'm new to this and I'm learning the basics now. In addition, i have noticed someone who suggested to start with the characteristic function.Can you give the way to do it using the MGF?Ray Vickson said:Why not use the moment-generating function instead?
deema_master said:I'm new to this and I'm learning the basics now. In addition, i have noticed someone who suggested to start with the characteristic function.Can you give the way to do it using the MGF?
Okay..but after we find the characteristic function or the MGF of their product...what does the resulted formula indicates? That's what i need to know...how is this formula related to the pdf of the sum of these random variables?Ray Vickson said:The MGF of a sum of independent RVs is the product of their MGFs---the same as it is for characteristic functions or Laplace transforms.
deema_master said:Okay..but after we find the characteristic function or the MGF of their product...what does the resulted formula indicates? That's what i need to know...how is this formula related to the pdf of the sum of these random variables?
I started from here https://www.physicsforums.com/threads/sum-of-noncentral-chi-square-rvs.374004/ , a homework helper "statdad" has suggested the method and i followed him, i did find the CF of the multiplication of the RV's CF, i just need to know how to continue.Ray Vickson said:I suggest you answer that question for yourself, by first writing out the details for the sum of 2 iid ##X##s.
Another way to solve the problem is to go right back to the beginning: what is the random variable that has the non-central chi-squared distribution? How is it related to the normal distribution? Answering that allows a simple solution almost by inspection and without any calculations at all!
deema_master said:I started from here https://www.physicsforums.com/threads/sum-of-noncentral-chi-square-rvs.374004/ , a homework helper "statdad" has suggested the method and i followed him, i did find the CF of the multiplication of the RV's CF, i just need to know how to continue.