A textbook defines the generating function f of a random variable X as:
Can anybody please explain that to me?
So far I know:
Generating function of random variable X can also be defined as:
So if we do differentiation:
How is this related to the very first equation?
Any help is much appreciated.
Lemme try to reconstruct my understanding after your inputs.
X is random variable 0,1,2,3,... with corresponding probability densities (which all add to 1).
We define the generating function of X as:
Turning it around, we can now see a new random variable (this is the key part which i couldn't see before), and the generating function is also the Expectation of this new variable,
Of course, this is not to be confused with E(X), which is derived from the differentiation of the 1st equation, and equating .
Am I correct?