Johnny, a sixth grader at Honey Creek Middle School in Terre Haute, Indiana, is required to sell candy bars in his neighborhood to raise money for the 6th grade field trip. There are thirty homes in his neighborhood, and his father has told him not to return home until he has sold five candy bars. So the boy goes door to door, selling candy bars. At each home he visits, he has an 0.4 probability of selling one candy bar and an 0.6 probability of selling nothing.
1  sum(f(j), j=5..30) = 1  .9985 = .0015
Moral: Negative binomial distributions don't turn our children out on the streets; bad parenting does.
If X_{r} is a random variable following the negative binomial distribution with parameters r and p, then X_{r} is a sum of r independent variables following the geometric distribution with parameter p. As a result of the central limit theorem, X_{r} is therefore approximately normal for sufficiently large r.
Furthermore, if Y_{s} is a random variable following the binomial distribution with parameters s and p, then
The negative binomial distribution also arises as a continuous mixture of Poisson distributions for which the Poisson parameter λ was generated by a Gamma distribution.
Suppose X is a random variable with a negative binomial distribution with parameters r and p. The statement that the sum from x=r to infinity, of the probability Pr[X = x], is equal to 1, can be shown by a bit of algebra to be equivalent to the statement that (1p)^{r} is what Newton's binomial theorem says it should be.
Suppose Y is a random variable with a binomial distribution with parameters n and p. The statement that the sum from y=0 to n, of the probability Pr[Y = y], is equal to 1, says that that 1 = (p + (1p))^{n} is what the strictly finitary binomial theorem of highschool algebra says it should be.
Thus the negative binomial distribution bears the same relationship to the negativeintegerexponent case of the binomial theorem that the binomial distribution bears to the positiveintegerexponent case.
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