for
a continuous distribution as
and
the moment generating function of a random variable
is
They
are called moment generating functions because we can obtain the
moments of a distribution from them.
A Level Maths Notes: S4 – Summary of Moment Generating Functions
The moment generating
function for a discrete distribution is defined asfor
a continuous distribution asand
the moment generating function of a random variableisThey
are called moment generating functions because we can obtain the
moments of a distribution from them.
Definition
The
moment
of a discrete random variable with probability mass function
and
is
The
first moment is the mean
corresponds
to the centre of mass in mechanics, whose position is found by taking
moments 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
is
the
moment.
Example: Find the moment generating function of the exponential distribution.
The probability density function is
The moment generating function gives rise to a method of finding the variance, of which more later.