While finding the mgf of the binomial distribution we allow for the X to cover all values from 0 to N, the number we "choose". Why don't we have to cover all possible values for the p and q as well?
While finding the mgf for the exponential distribution, we set X as the number we multiply by the rate parameter. Why isn't X over all possible rate parameters instead?
While finding the mgf for the normal distribution, we set X as all possible values on the X curve. Why can we assume it works for all means and standard deviations? Why doesn't X stand for the standard deviation?
I apologize in advance that this is a "duh" question; I feel that I am missing a fundamental intuition and I'd much rather understand it than memorize it.
Thank you.
Anthony