GX(t)=E(tX)=∑P(X=x)txG_X(t)=E(t^X)=\sum P(X=x)t^xGX(t)=E(tX)=∑P(X=x)tx packs a whole distribution into one function. G(1)=1G(1)=1G(1)=1, E(X)=G′(1)E(X)=G'(1)E(X)=G′(1) and Var(X)=G′′(1)+G′(1)−[G′(1)]2\text{Var}(X)=G''(1)+G'(1)-[G'(1)]^2Var(X)=G′′(1)+G′(1)−[G′(1)]2. For independent XXX and YYY, GX+Y=GXGYG_{X+Y}=G_XG_YGX+Y=GXGY.