The moment generating function of a random variable is another way of expressing its probability distribution and means many analytic results can be derived via another route. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables.
The moment-generating function of a random variable X is
wherever this expectation exists.
always exists and is equal to 1.
Moments and the moment-generating function may not exist, as the integrals need not converge. By contrast, the characteristic function always exists (because it is the integral of a bounded function on a space of finite measure), and thus may be used instead.
More generally, where
an
- dimensional random vector, one uses
instead of
The moment generating function can be used to find all the moments of the distribution. The series expansion of
is
where
is the ith moment.
If we differentiate
times with respect to
and then set
we obtain the ith moment,
Some moment generating functions are shown below.
|
Distribution |
Moment-generating function MX(t) |
|---|---|
|
Binomial |
|
|
Poisson |
|
|
Uniform |
|
|
Normal |
|
|
Chi-square |
|
|
Gamma |
|
|
Exponential |
|
|
Laplace |
|
|
Negative Binomial NB(r, p) |
|
The moment-generating function is so called because if it exists on an open interval around
then it is the exponential generating function of the moments of the probability distribution:
