Probability generating function question

[minorqueen]

I'm starting to go thru the Klugman book, Loss Models. On page 201 he defines a probability generating function. I have never seen this before. Can anyone explain how this is related to the moment generating function?

[Apollywog]

I never actually read the text book but I think it's just saying that the probability generating function can be used to find moments (while the moment generating function can do the same).

Correct me if this wasn't what the line under Definition 3.1 means.

[Black Mage]

I think that intuitively the moment generating function can be used to find the aggregate [continuous] probability that something occurs. The mgf is nice since, when independent, exponents from E[e^xt] can be added to get an overall probability.
The pgf, probability generating function is a discrete way to show aggregate probability.[/b]

[G.V.Ramanathan]

For any sequence, {a_n} you can define a generating function by
P(z) = Σ z^n a_n . (It helps if the series converges in some open interval including 0 .) Then it implies that the n-th term in the Taylor series around the origin of P(z) has coefficient a_n. This is a neat way of representing the sequence. It is very useful in a number of contexts including the solution of difference equations. In particular, if X is a discrete random variable with distribution Pr(X = n) = a_n, then P(z) is called the Probability Generating Function (PGF). In that case P(z) = E(z^X). If you wish to learn more about it I recommend William Feller’s book titled, “An Introduction to Probability Theory and Applications,” Volume 1.

The Moment Generating Function (MGF) is defined as
M(t) = E(e^tX) = E(1 + tX + t^2X^2/2 + …). So the coefficient of (t^n)/n! is the n-th moment of X.

Clearly, M(t) = P(e^t) and P(z) = M(ln z). Basically, the power series for PGF “generates” probabilities and the power series for the MGF “generates” moments. The k-th derivative of the MGF at 0 is the k-th moment of X and the k-th derivative of the PGF at 0 is k! times Pr(X=k). (There are some subtleties which we ignore).

[minorqueen]

Thank you everyone.

[Summer]

If you are still having a problem, it is explained in much more detail in course 1 material. You may benefit from looking back and refreshing your memory.