Sums of two probability density functions

Question

How to find the sum of two probability density functions?What is the formula used for this purpose?

Answer 1

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.

Answer 2

As learner points out, its the convolution you are after. Better context on what the individual probability density functions are would help.