pdf of sum of two dependent random variables What is the pdf of sum of two dependent random variables given we know their joint
pdf and individual pdfs. I have seen already some posts but none of them answered when they are dependent. Every one solved for only the independent case but i need for dependent case in terms of the joint pdf and individual pdfs in an explicit form.
If $f(x,y)$ denotes the PDF of $(X,Y)$ and $Z=X+Y$ then:
$$f_Z(z)=\int^{\infty}_{-\infty}f(x,z-x)dx$$

Substituting $y=u-x$ we find:$$F_Z(z)=\int\int_{x+y\leq z}f(x,y)dxdy=\int^{\infty}_{-\infty}\left[\int^{z-x}_{-\infty}f(x,y)dy\right]dx=$$$$\int^{\infty}_{-\infty}\left[\int^{z}_{-\infty}f(x,u-x)du\right]dx=\int^{z}_{-\infty}\left[\int^{\infty}_{-\infty}f(x,u-x)dx\right]du$$

Taking the derivative of $F_Z$ we come to the mentioned result.
Let $X,Y$ be dependant variables with PDF $f(x,y)$.

Then sum PDF is: $$f_{X+Y}(k)= \int f(x,k-x) dx$$