Laplace transform of noncentral chi-square distribution

Question

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function.

Let me know whether it exists or relevent book and paper.

Thanks in advance

Answer 1

If you mean the probability function for the chi-square distribution with $r$ degrees of freedom $$P_r(x)=\frac{x^{-1+r/2}e^{-x/2}}{2^{r/2}\Gamma(r/2)}$$ The Laplace transform is : $$\int_0^\infty P_r(x)e^{-sx}dx = \frac{(s+1/2)^{-r/2}\Gamma(r/2)}{2^{r/2}\Gamma(r/2)}= (2s+1)^{-r/2}$$

Answer 2

Using Maple, I find the Laplace transform of the noncentral chi-square density with $\nu$ degrees of freedom and noncentrality parameter $\delta$ is

$$ \left( 1+2\,s \right) ^{-\nu/2} \exp\left(-{\frac {\delta\,s}{1+2\,s}}\right) $$

Answer 3

It is mentioned in the following paper: http://link.springer.com/article/10.1007/BF02595410