Probability Generating Functions

1
Probability Generating Functions

• Suppose a RV X can take on values in the set of
  non-negative integers 0, 1, 2, .
• Definition The probability generating function
  of X is

2
Probability Generating Functions

• In general, the p.g.f. of the sum of independent
  RVs is the product of the p.g.f.s.

3
Geometric Distribution

• The waiting time for the first success in a
  sequence of Bernoulli trials (with success
  probability p) has a geometric distribution on
  1, 2, with parameter p.
• Let T the waiting time
• P(T i) qi?1p (i 1, 2, )

4
Negative Binomial Distribution

• The waiting time for the rth success in a
  sequence of Bernoulli trials (with success
  probability p) has a negative binomial
  distribution on r, r1, r2, with parameters
  r and p.
• Let Tr the waiting time for rth success

5
Exercise

• If a child exposed to a contagious disease has a
  40 chance of catching it, what is the
  probability that the tenth child exposed is the
  third to catch the disease?

6
Poisson Distribution

• As we saw in Section 2.4, the Poisson
  distribution is a good approximation to the
  binomial distribution when n is large and p is
  small. The Poisson distribution has other
  interesting applications.
• N has a Poisson distribution with parameter m if