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
Famous quotes containing the words important and/or properties:
“The reason people think it’s important to be white is that they think it’s important not to be black.”
—James Baldwin (1924–1987)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)