The moment generating function (MGF) of a r.v. is a function of . It’s defined as .

Note that whatever we choose for might cause the expectation to blow up (approach ). This is fine, as long as is finite for all in some open interval around zero. Otherwise, the MGF of does not exist.

Here, is a random variable that depends on and the original random variable .

If the MGF exists, then when then . There’s no intuitive explanation for what really is. LOL.


There are three theorems involving MGFs that we’ll use.

  1. If exists, then equals the th derivative of the MGF, evaluated at zero. That is, we have for

For a not very rigorous proof, we can consider the Taylor series, replace the in with , and see that every term corresponds to a derivative of the moment.

  1. If and have the same MGF, then they must also have the same distribution.
  2. If and are Independent and both of their MGFs exist, then the MGF of is the product of their MGFs. We have:

Location-scale transformation

If has a MGF of , then the MGF of is given by

Since is a constant, we can basically pretend that we’re transforming . If we let , then the above is the same as , which is the definition of the MGF.

This can be useful for finding the MGF of a general continuous distribution after we derive the MGF of the standard version of the distribution (e.g. uniform, exponential, Normal).

Solving the MGF for a distribution

When we finish deriving a MGF for a distribution, we can case on what values can take on, and see where we define our neighborhood .

  • We can pick the cutoff so that is well defined.
  • Remember, as long as such an interval exists, then the MGF of can exist.

Discrete variable

To solve the MGF, begin with the definition of expectation. We can use LOTUS because is really a transformation of , with .

where denotes us iterating over all the possible values of the random variable . We substitute with whatever relevant PMF.

Continuous variable

Instead of a summation, we use an integral instead over . Remember that the PDF is often nonzero outside of a specific range, so the integral simplifies. We’re also using LOTUS here as well.