## Summary of Moment Generating Functions

The moment generatingfunction for a discrete distribution is defined asfora continuous distribution asandthe moment generating function of a random variableisTheyare called moment generating functions because we can obtain themoments of a distribution from them.

Definition

Themomentof a discrete random variable with probability mass functionandis

Thefirst moment is the meancorrespondsto 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.

Thenwhereisthemoment.

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.