If the MGF is defined for an interval of values of , then

so every moment of is read off from a derivative of at . In particular and .

If are independent random variables, then

so the MGF of a sum of independent random variables is the product of their MGFs.

For example, if and are independent, then for ,

so ; in particular, the sum of i.i.d. random variables has the distribution.

For with density , completing the square in the exponent gives

The first term integrates against the density of to , so

Together with multiplicativity under sums this recovers for independent normals and .

For the multivariate MGF is

If it is finite for an open set of values of , it uniquely determines the distribution of , and partial derivatives at the origin give moments such as and . Moreover,

if and only if are independent.

The Cauchy distribution has density

and its MGF is

Hence all have the same MGF while having different distributions. The finiteness of the MGF on an open interval of values of is therefore a necessary hypothesis in the uniqueness theorem.