Random Sums
If is independent of the i.i.d. sequence and , the PGF of is the composition
where and .
Lemma 3.65
Let be the PGF of and be the PGF of .
Then the PGF of is given by
Proof
Conditioning on the value of ,
Equivalently, with the tower property,
and independence of from gives , so and .
Mean and variance
Differentiating at and using ,
Related
Stated in
- Lemma 3.65§3.6.2 Sum of a Random Number of Random Variables