Sums of two probability density functions If the weighted sum of 2 probability density functions is also a probability density function, then what is the relationship between the random variables of these 3 probability density functions.

I think you mean how to find the probability density of the random variable that is the sum of two other random variables, using the probability densities of these two variables. The answer is that the probability density of the sum is the convolution of the densities of the two other random variables if they are independent.
Let's say $Z = X + Y$, then the density of the sum is given by
$$ f_Z \left( z \right) = \int_{-\infty}^{\infty} f_X \left( z - y \right) f_Y \left( y \right)
   d y $$
assuming all variables are real valued, that $X,Y$ are independent and that $f_X,f_Y,f_Z$ are the densities of $X,Y,Z$ respectivley.
