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 ).
Oh right sorry, I misunderstood your question.
There is a big mistake for the third '=' sign :
, which is completely different from what you wrote.
I would like to know what you mean by "the beginning" and "the end". Do you mean that the book only provides ?