The cumulant generating function takes the use of MGF, Mx(t), one step further. It is primarily useful in finding the expected value and variance of a random variable. The cumulant generating function of X is denoted as Kx(t). It is defined as the natural logarithm of the MGF of X. In math notations, the relationship is as follows:

Kx(t) = ln [Mx(t)]

Kx(t) has a few useful properties, but the two I’m about to derive are the most relevant.

Property #1: The first derivative of Kx(t) evaluated at t=0 gives us E(X). Refer to derivation #1 in the attached file.

Property #2: The second derivation of Kx(t) evaluated at t=0 gives us Var(X). Refer to derivation #2 in the attached file.

Kx(t) is most helpful when the MGF is expressed as an exponential function, and when asked to calculate the variance. Refer to example #1 in the attached file where I found the Var(X) using Kx(t) and also Mx(t).

In general, when given the MGF of a random variable, we can use Kx(t) to find E(X) and Var(X) instead of making use of the MGF directly. As can be seen in the attached file, using the MGF to find the Var(X) was a much longer method than just using Kx(t) so yes, it sometimes can be used as a shortcut depending on how the MGF is expressed.