# numerically solve summation of random variables integral

I want to add two random variables, $$X,Y$$ together $$Z=X+Y$$. They may be correlated. The integral is $$f_z(z)=\int^{\infty}_{-\infty}\int^{z-x}_{-\infty}f_{x,y}(x,y)dxdy=\int_{-\infty}^{\infty}f_{x,y}(x,z-x)dx$$ where $$f_z$$ is the pdf of the new random variable $$Z$$ and $$f_{x,y}$$ is the joint pdf of the random variables $$x,y$$.

My problem is that I have two marginal distributions $$f_x(x)$$ and $$f_y(y)$$. I can estimate the joint pdf using a copula.

However, this is all done numerically. I have two arrays for $$x$$ and $$y$$, and two arrays for $$f_x(x)$$ and $$f_y(y)$$.

I can calculate the joint pdf using the copula method, but again it is numerical.

I wrote some code to test this using Gaussians since the solution is known and I can easily check if I am correct.

I first calculated the joint pdf in $$x,y$$. Then I created the Z array, $$z=x+y$$. And I integrated the joint pdf, which is a 2-d matrix, with respect to x and plotted it along the z array. But integrating along x is produces the same answer if I was still using a joint pdf in x and y. So this gave me a wrong answer.

I then tried to calculate the joint pdf in $$x,z$$ by converting the marginal distribution of $$y$$ to $$x,z$$. But this didn't work.

I finally tried to do the double integral and use the upper limit of $$z-x$$. But I couldn't get this to work correctly either.

It is possible that I have some bugs in my code and one of these solutions will work correctly.

It seems that this should be doable because if $$f_{x,y}(x,z-x)=f_{x,z}(x,z)$$ the 2d matrix I have should be the same. But maybe i'm wrong because z is a continuous variable, and those two functions are only equal if $$z=x+y$$.