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.
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:
$\displaystyle P\left(\left.\left(\chi '\right)^2\right|\nu ,\lambda \right)\to P(x)$
$\displaystyle x\text{:=}\sqrt{\frac{2 \left(\chi '\right)^2}{b+1}}-\sqrt{\frac{2 a}{b+1}-1}$
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