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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.

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Probability Generating Functions

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  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

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