Suppose that { } is a sequence of intedependent identically distributed random variables and that N is a random variable taking non-negative integer values.

If , then

Proof.

The beginning and the end is from the textbook, the part in the middle with is mine.

I am not quite sure about the last transition

.

I can see how becomes . I am not sure why if there is no expected value E anywhere around (since the definition of MGF is ).