Definition 2.8. Let be a random variable and let be independent identically distributed random variables that are independent of , too.

The random variable is called a random sum (, ).

The distribution of can be obtained by using the theorem of total probability. Similarly, the moments of the random sum can be calculated by the help of the theorem of total moments.

Discrete case

Continuous case

Example 2.13. Let , and let be geometrically distributed with parameter . Find the density function of .

Solution:

Notice that is Erlang distributed with parameters hence by substituting its density function we have

It means that .

Theorem 2.9. Mean of a random sum

Proof. By the law of total expectation we have

Theorem 2.10. Variance of a random sum

Proof. By applying the theoerm of total moment we get

Thus