*Z*=*X*+*Y*

Since the events are disjoint we just sum their probabilities:

Definition:
For X and Y two random variables
with mass distribution functions *m*_{1} and *m*_{2}, the convolution
of *m*_{1} and *m*_{2} is the distribution function
*m*_{3} denoted *m*_{1}**m*_{2} given by:

This operation is commutative and associative.
We use induction to find the distribution
of the sum of *n* random variables.
Suppose that *S*_{k} is the sum of *k* dice:

See another example for the distribution of the sum of points in a bridge hand.