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.
A non-central chi-square distribution is a probability distribution that is used to model the sum of squares of independent random variables. It is similar to a chi-square distribution, but it takes into account a non-centrality parameter, which represents the mean of the distribution. This distribution is often used in statistical analysis to test hypotheses and make inferences about population parameters.
The PDF of the sum of n, iid, non-central chi-square is calculated by taking the product of the individual PDFs of the non-central chi-square distributions and then integrating over all possible values of the random variables. This can be a complex calculation, but it can be simplified for certain cases, such as when the non-centrality parameter is equal to zero.
The non-centrality parameter in a non-central chi-square distribution represents the mean of the distribution. It determines the location of the peak of the distribution and affects its shape. A larger non-centrality parameter leads to a distribution with a higher peak and a longer tail, while a smaller non-centrality parameter results in a flatter distribution.
The non-centrality parameter and the degrees of freedom in a non-central chi-square distribution are both important parameters that affect the shape of the distribution. As the degrees of freedom increase, the distribution becomes more symmetrical and approaches a normal distribution. On the other hand, as the non-centrality parameter increases, the distribution becomes more skewed and has a longer tail.
The non-central chi-square distribution has many applications in statistical analysis, such as in hypothesis testing and confidence interval estimation. It is commonly used in fields such as economics, engineering, and physics to model and analyze data. Additionally, it is also used in Monte Carlo simulations and in the design of experiments.