Probability Generating Function

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Probability Generating Function The Probability Generating Function (PGF) is similar to the Moment Generating Function, but (1) it is usually only discussed for discrete random variables, (2) it is much less useful, in my opinion, and (3) you are much less likely to encounter it on the exam. I am conﬁdent I did not study a single Probability Generating Function when I took exam P. That said, it’s in the study manual, so let’s get to learning. The Probability Generating Function is represented with a capital P , which we almost always use to represent Probability of an event. As such, it will be speciﬁed when it refers to the PGF. You may assume P represents the PGF for this entire section. Here is our main formula:

N • PN (z) = E[z ]

The probability generating function generates two different things. The first is factorial moments, which are th obtained by evaluating derivatives of the PGF at z = 1. The n factorial moment, µ(n) is defined by

• µ(n) = E[X(X − 1)(X − 2) ··· (X − n + 1)]

To clarify,

0 000 • P (1) = µ(1) = E[X] • P (1) = µ3 = E[X(X − 1)(X − 2)]

00 0000 • P (1) = µ(2) = E[X(X − 1)] • P (1) = µ4 = E[X(X − 1)(X − 2)(X − 3)] and in general,

(n) • P = µ(n) = E[X(X − 1)(X − 2) ··· (X − n + 1)]

Is this useful? I don’t really know. You can use some algebra to turn factorial moments into regular moments (that is, E[Xn]), but there are easier ways to do that. The other function of PGFs is more aligned with its name. A PGF can generate probabilities by evaluating derivatives at z = 0. Let pn be the probability that a non-negative, discrete random variable is equal to (n) n, that is, pn = P r(N = n). In the following formula, note that pn represents a probability, while P represents a derivative of the PGF. Then, P 00(0) • P r(N = 0) = p0 = P (0) • P r(N = 2) = p = 2 2! P 000(0) • P r(N = 1) = p = P 0(0) • P r(N = 3) = p3 = 1 3! and in general,

P (n)(0) • P r(N = n) = p = n n!

The Probability Generating Function can be related to the Moment Generating Function with the formula

• PN (z) = MN (ln z)

Lastly, for independent RVs,

• PX+Y (z) = PX (z)PY (z)

1 Ex. The discrete random variable N has the following probability function:

  0.5 n = 0    0.3 n = 1 P (N + n) =  0.2 n = 2    0 otherwise

Calculate PN (10), the probability generating function of N evaluated at 10.

2 Ex. A discrete random variable has probability generating function PN (z) = (1 + 0.2(z − 1)) . Calculate the moment generating function of this random variable evaluated at 1.

2 The probability generating function for a random variable X is

4 6 PX (z) = 0.28 + 0.37z + 0.23z + 0.12z

Calculate the coeﬃcient of variation of X.

Ex. For a random variable, X, let M(z) be the Moment generating function and P (z) be the probability generating function. You are given:

i) M 0(0) = 5.

ii) P 00(1) = 35.

Determine Var(X).