# What is the difference between moment generating function and probability generating function?

I am confused between the two terms ” probability generating function” and “moment generating function.” How do those terms differ?

The probability generating function is usually used for (nonnegative) integer valued random variables, but is really only a repackaging of the moment generating function. So the two contains the same information.

Let $X$ be a non-negative random variable. Then (see https://en.wikipedia.org/wiki/Probability-generating_function) the probability generating function is defined as

and the moment generating function is

Now define $\log z=t$ so that $e^t=z$. Then

So, to conclude, the relationship is simple:

EDIT


@Carl writes in a comment about this my formula ” … which is true, except when it is false” so I need to have some comments. Of course, the equality $G(z) = M_X(\log z)$ assumes that both are defined, and a domain for the variable $z$ need be given. I thought the post was clear enough without that formalities, but yes, sometimes I am too informal. But there is another point: yes, the probability generating function is mostly used for (nonnegative argument) probability mass functions, wherefrom the name comes. But there is nothing in the definition which assumes this, it can as well be used for any nonnegative random variable! As an example, take the exponential distribution with rate 1, we can calculate

which could be used for all purposes we do use the moment generating function, and you can check the relationships between the two function are fulfilled. Normally we do not do this, it is probably more practical to use the same definitions with (possibly) negative as well as with nonnegative variables. But it is not forced by the mathematics.