Claims on a group of policies of a certain type arise as a Poisson process

with parameter ${}_1=1.$ Claims on a second, independent, group of

policies arise as a Poisson process with parameter ${}_2=2.$ The

aggregate claim amounts on the respective groups are denoted $S_1,S_2.$

Using moment generating functions $($or otherwise$),$ show that

$S_1+S_2)$ also has a compound Poisson distribution and hence derive

the Poisson parameter for $S.$