Chapter 5.4, Problem 9E

Let X and Y, with respective pmfs f(x) and g(y), be independent discrete random variables, each of whose support is a subset of the nonnegative integers 0, 1, 2,… Show that the pmf of $W=X+Y$ is given by the convolution formula $h\left(w\right)={\displaystyle \underset{x=0}{\overset{w}{\sum }}f\left(x\right)g\left(w-x\right)},w=0,1,2,\mathrm{....}$

Hint: Argue that $h\left(w\right)=P\left(W=w\right)$ is the probability of the $w+1$ mutually exclusive events $\left\{X=x,Y=w-x\right\},x\in \left\{0,1,\mathrm{...},w\right\}$.