The moment generatingfunction for a discrete distribution is defined as
fora continuous distribution as
andthe moment generating function of a random variable
is
Theyare called moment generating functions because we can obtain themoments of a distribution from them.
Definition
The
momentof a discrete random variable with probability mass function
andis
Thefirst moment is the mean
correspondsto the centre of mass in mechanics, whose position is found by takingmoments about some points or axes.
The moments can be generated using the Taylor series expansion of
We can write
Multiply out the brackets and integrate each term separately.
Then
where
isthe
moment.
Example: Find the moment generating function of the exponentialdistribution.
The probability density function is
The moment generating function gives rise to a method of findingthe variance, of which more later.