Probability-generating Function - Examples

Examples

  • The probability-generating function of a constant random variable, i.e. one with Pr(X = c) = 1, is
  • The probability-generating function of a binomial random variable, the number of successes in n trials, with probability p of success in each trial, is
Note that this is the n-fold product of the probability-generating function of a Bernoulli random variable with parameter p.
  • The probability-generating function of a negative binomial random variable, the number of trials until the rth success with probability of success in each trial p, is
(Convergence for ).
Note that this is the r-fold product of the probability generating function of a geometric random variable.
  • The probability-generating function of a Poisson random variable with rate parameter λ is


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