Density of a sum of independent variables is given by convolution
This topic is pretty complex because it involves properties of integrals that economists usually don't study. I provide this result to be able to solve one of UoL problems.
General relationship between densities
Let be two independent variables with densities . Denote the joint density of the pair .
By independence we have
These are the only simple facts in this derivation. By definition,
For the last probability in (3) we have a double integral
Using (1), we replace the joint probability by the product of individual probabilities and the double integral by the repeated one:
The geometry is explained in Figure 1. The area is limited by the line . In the repeated integral, we integrate first over red lines from to and then in the outer integral over all .
(3) and (4) imply
Finally, using (2) we differentiate both sides to get
This is the result. The integral on the right is called a convolution of functions .
Remark. Existence of density (2) follows from existence of , although we don't prove this fact.
Exercise. Convolution is usually denoted by . Prove that
- If is uniformly distributed on some segment, then is zero for large .