I am looking for the closed-form expression of the CDF of the product of two independent generalized non-central chi distributions (not chi-squared) each with k=2 degrees of freedom. A generalized non-central chi distribution (with k=2) is the square root of the sum of squares of two Gaussian random variables of different means and variances.
I came across two posts on StackExchange where this problem has been discussed before.
This post mentions that a closed-form expression of the pdf of the product of two independent non-central chi distribution with k=2 has been derived before. Since the variances of the constituting Gaussian variables are identical in the paper mentioned at this link, this is not a generalized non-central chi distribution.
This post states that for the generalized non-central chi-squared distribution, a closed-form pdf or cdf is not available at least for arbitrary degrees of freedom. As a result, a product of two generalized non-central chi-squared is also not available.
I am looking for pdf and cdf of generalized non-central chi (not chi-squared) with only k=2 degrees of freedom. I am guessing that this might be easier compared to deriving an expression with arbitrary degrees of freedom. Does such an expression exist?