1 / 6

Probability Generating Functions

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. Probability Generating Functions.

meda
Download Presentation

Probability Generating Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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, r+1, r+2, …} 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

More Related