Negative binomial distribution
The negative binomial distribution is a discrete probability distribution that describes the probability that a Bernoulli event with probability p of success has occurred a given number of times after a given number of trials. If the random variable X represents the number of trials to get the r-th success, then X follows the negative binomial distribution with parameters r and p.
Formulas
- Parameters
- r (number of successes) is an integer where 1 ≤ r; the special case r = 1 creates the geometric distribution. : p (probability of success on any given trial) is a real number where 0 < p < 1.
- Support (domain where probability mass > 0)
- integers ≥ r
- Probability mass function f(x) = P(X = x) (probability that rth success occurs on trial x)
- C(x - 1, r - 1) pr(1 - p)x - r (see binomial coefficient)
- Cumulative distribution function F(x) = P(X ≤ x) (probability that r-th success occurs on or before trial x)
- No closed form solution; just use sum(f(j), j = r..floor(x))
- Expected value E[X]
- r/p
- Variance σ2
- (r - rp)/p2
Example
after a problem by Dr. Diane Evans (a math professor at Rose-Hulman Institute of Technology)
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.
What's the probability mass function for selling the last candy bar at the xth house?
- f(x) = .01024*((x - 1) choose 4)*.6^(x - 5)
What's the probability that he finishes on the tenth house?
- f(10) = .100
What's the probability that he finishes on or before reaching the eighth house?
Answer: To finish on or before the eighth house, he must finish at the fifth, sixth, or seventh house. Sum those probabilities:
- f(5) = 0.102; f(6) = .0307, f(7) = .0553; sum(f(j), j=5..7) = 0.963
What's the probability that he exhausts all houses in the neighborhood, gives up, and then goes to live on the streets?
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.
Properties
If Xr is a random variable following the negative binomial distribution with parameters r and p, then Xr is a sum of r independent variables following the geometric distribution with parameter p. As a result of the central limit theorem, Xr is therefore approximately normal for sufficiently large r.
Furthermore, if Ys is a random variable following the binomial distribution with parameters s and p, then
- Pr[Xr ≤ s] = Pr[Ys ≥ r] = Pr["after s trials, there are at least r successes"]
In this sense, the negative binomial distribution is the "inverse" of the binomial distribution, hence the name. Every statement about negative binomial distributions can be reformulated into an equivalent statement about binomial distributions.
FED