Approximations to the Negative Binomial distribution

The Negative Binomial distribution NegBin(s,p) models the number of failures 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:

NegBin(s, p) » Normal(, )

The approximation can be justified via Central Limit Theorem, because the NegBin(s, p) distribution can be thought of as the sum of s independent NegBin(1, p) distributions, each with mean and standard deviation .

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 NegBin(1,p) distribution:

As p gets very small, the NegBin(1,p) becomes an Exponential distribution (see below), and so we can use a Gamma approximation to the NegBin instead of a Normal. On the other hand, as p is large, so the NegBin(1,p) 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:

NegBin(s,0.5) distributions and their corresponding Normal distribution approximations

NegBin(s,0.9) 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(a,b) distribution models the 'time' until observing a events where b is the mean time between events. The NegBin distribution is the binomial equivalent, modeling the number of failures to achieve s successes where [(1/p)-1] is the mean number of failures per success. The NegBin excludes 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 exactly comparable, we should therefore think of the mean number of trials per success, equal to (1/p). Then, we can make the following approximation:

NegBin(s,p) » Gamma(s,1/p)                           when      p » 0

For s = 1, we also have the special case:

Geometric(p) » Expon(1/p)                              when      p » 0

When the Exponential distribution is a good approximation to the Geometric (p<0.05 is usually good, see below), the Gamma is a good approximation to the NegBin.

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