The probability generating function is a power series that gives us the probabilities associated with a discrete random variable.
Definition:
Let X be a discrete random variable that takes non-negative values 0, 1, 2,….etc. Let pi=P(X=i) denote the probability that the random variable X takes the value ‘i’. Then the probability generating function (p.g.f) is a power series in ‘s’ (denoted P(s)) defined as,
If we differentiate the above power series term by term ‘n’ times we get,
Substituting s=0 on both sides and rearranging terms we get,
Thus we see that we can obtain the probabilities associated with a random variable by differentiating the probability generating function.
Examples of Probability-Generating Functions:
1. The probability generating function for the Poisson distribution with parameter λ is given as:
P(s) = e-λesλ.
2. The probability generating function for the geometric distribution with parameter p is given as:
P(s) = p/(1-qs).
3. The probability generating function for the binomial distribution with parameters n and p is given as:
P(s) = (q+ps)n.
4. The probability generating function for the negative binomial distribution with parameters r and p is given as:
P(s) = (p/(1-qs))r.
Properties of P.G.F:
- Let X and Y be two independent discrete random variables. Then the p.g.f of X+Y is equal to the product of their respective p.g.f’s, that is, P{X+Y}(s) = PX(s)PY(s).
- If the p.g.f is differentiated ‘r’ number of times, then on substituting s=1, we obtain the rth factorial moment.
- If we substitute s=et in the p.g.f then we obtain the moment generating function of our random variable.