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Moment Generating Functions

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 usesinstead of

The moment generating function can be used to find all the moments of the distribution. The series expansion ofis

whereis the ith moment.

If we differentiatetimes with respect toand then setwe 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: