# Thread: help me to convolute non central chi-square with white Gaussian distribution

1. ## help me to convolute non central chi-square with white Gaussian distribution

Hi, I am doing research and i am stuck at this point I need help to convolute non central chi-square with white Gaussian distribution.

2. ## Re: help me to convolute non central chi-square with white Gaussian distribution

Do you need approximation or exact formulas?

3. ## Re: help me to convolute non central chi-square with white Gaussian distribution

Thanks for your reply I really appreciate your help.......approximation will be OK if you don't know the exact formula....

4. ## Re: help me to convolute non central chi-square with white Gaussian distribution

Non-Central $\displaystyle \chi ^2$ Distribution in terms of $\displaystyle \chi ^2$ Distribution:

$\displaystyle P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\text{:=}\sum _{j=0}^{\infty } \frac{e^{-\frac{\lambda }{2}} \left(\frac{\lambda }{2}\right)^j P\left(\left.\left(\chi '\right)^2\right|2 j+\nu \right)}{j!}$

$\displaystyle \lambda \geq 0$ non-centrality parameter.

Non-Central $\displaystyle \chi ^2$ Distribution in terms of Normal Distribution:

$\displaystyle \to$ means approximately equal.

$\displaystyle P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\to P(x)$

$\displaystyle x\text{:=}\frac{\sqrt[3]{\frac{\left(\chi '\right)^2}{a}}-\left(1-\frac{2 (b+1)}{9 a}\right)}{\sqrt{\frac{2 (b+1)}{9 a}}}$

A simpler approximation:

$\displaystyle P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\to P(x)$

$\displaystyle x\text{:=}\sqrt{\frac{2 \left(\chi '\right)^2}{b+1}}-\sqrt{\frac{2 a}{b+1}-1}$

5. ## Re: help me to convolute non central chi-square with white Gaussian distribution

Thank you for your help.....can I know what is end result of the convolution.

6. ## Re: help me to convolute non central chi-square with white Gaussian distribution

In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form

$\displaystyle \sum_{i=1}^n X_i \sim Y$

where $\displaystyle X_1, X_2,\dots, X_n\$, are independent and identically distributed random variables. In place of X_i and Y the names of the corresponding distributions and their parameters have been indicated.

List of convolutions of probability distributions - Wikipedia, the free encyclopedia