Separation of two probabilistic distribution: gaussian and exponential

I have a dataset where I highly suspect some correlation by two variables, based on my understanding of the problem. I would have a linear correlation (sort of SVD decomposition or a plane such as $y=mx+p$ for the main axis separation) of these variables. Assuming that I find the right separation, I would expect the principal axis to follow a gaussian distribution (high noise) and the secondary axis to follow and decreasing exponential probability (or maybe a poisson distribution with $lambda$ close to 1).

My question is: how to test that and how would I extract the key parameters ($lambda$ for exponential distribution, $mu$ and $sigma$ for the gaussian distribution and more importantly, the $m$ and $p$ for the $y=mx+p$ plane separation?)