The Negative Binomial distribution NegBinomial(p,s) models the total number of trials (n trials = s successes plus n-sfailures ) it takes to achieve s successes, where each trial has the same probability of success p.
Normal approximation to the Negative Binomial
When the number of successes s required is large, and p is neither very small nor very large, the following approximation works pretty well:
NegBinomial(p, s) » Normal | \Big(\frac{s}{p},\sqrt{s(1-p)/p^{2}}\Big) |
The approximation can be justified via Central Limit Theorem, because the NegBinomial(p,s) distribution can be thought of as the sum of s independent NegBinomial(p, 1) distributions, each with mean \frac{1}{p} and standard deviation \sqrt{\frac{1-p}{p^{2}}}.
The difficulty lies in knowing whether, for a specific problem, the values for s and p fall within the bounds for which the Normal distribution is a good approximation. The smaller the value of p, the longer the tail of a NegBinomial(p,1) distribution:
As p gets very small, the NegBinomail(p,1) becomes an Exponential distribution (see below), and so we can use a Gamma approximation to the NegBinomial instead of a Normal. On the other hand, as p is large, so the NegBinomial(p,1) distribution gets more skewed, so s would need to be much larger for a Normal approximation (which has to overcome this skewness) to be appropriate:
NegBinomial(0.5,s) distributions and their corresponding Normal distribution approximations
NegBinomial(0.9,s) distributions and their corresponding Normal distribution approximations, showing that when p is large, s needs to be higher for the Normal approximation to work well.
Gamma approximation to the Negative Binomial
The Poisson process can be derived from the Binomial process by making n extremely large while p becomes very small, but within the constraint that np remains finite. In a Poisson process, the Gamma(0,b,a) distribution models the 'time' until observing a events where b is the mean time between events. The NegBinomial distribution is the binomial equivalent, modeling the total number of trials to achieve s successes where [(1/p)-1] is the mean number of failures per success. The NegBinomial in Crystal Ball includes the s successes which in terms of a Poisson process are not included in the waiting time because each event is assumed to be instantaneous. To make the two approaches more comparable, we subtract the (non-random) number of successes from the NegBinomial(p,s) distribution to obtain the number of failures only (i.e. shift the distribution s to the left). The remaining distribution models the number of failures, with mean (1/p-1) failures for each success. Then, we can make the following approximation:
NegBinomial(p,s) - s » Gamma(0,1/p-1,s) when p ® 0
Or equivalently, using the shift parameter for the Gamma distribution:
NegBinomial(p,s) » Gamma(s,1/p,s) when p ® 0
For s = 1, we also have the special case:
Geometric(p) -1 » Exponential(p/(1-p)) when p ® 0
When the Exponential distribution is a good approximation to the "Geometric(p) - 1" (p<0.05 is usually good, see below), the Gamma is a good approximation to the NegBinomial.
