Important Properties
The most important property of the moment-generating function is that if two distributions have the same moment-generating function, then they are identical at almost all points. That is, if for all values of t,
then
for all values of x (or equivalently X and Y have the same distribution). This statement is not equivalent to ``if two distributions have the same moments, then they are identical at all points", because in some cases the moments exist and yet the moment-generating function does not, because in some cases the limit
does not exist. This happens for the lognormal distribution.
Read more about this topic: Moment-generating Function
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