The negative binomial distribution has a natural intepretation as a waiting time until the arrival of the rth success (when the parameter r is a positive integer). The waiting time refers to the number of independent Bernoulli trials needed to reach the rth success. This interpretation of the negative binomial distribution gives us a good way of relating it to the binomial distribution. For example, if the rth success takes place after failed Bernoulli trials (for a total of trials), then there can be at most successes in the first trials. This tells us that the survival function of the negative binomial distribution is the cumulative distribution function (cdf) of a binomial distribution. In this post, we gives the details behind this observation. A previous post on the negative binomial distribution is found here.
A random experiment resulting in two distinct outcomes (success or failure) is called a Bernoulli trial (e.g. head or tail in a coin toss, whether or not the birthday of a customer is the first of January, whether an insurance claim is above or below a given threshold etc). Suppose a series of independent Bernoulli trials are performed until reaching the rth success where the probability of success in each trial is . Let be the number of failures before the occurrence of the rth success. The following is the probablity mass function of .
Be definition, the survival function and cdf of are:
For each positive integer , let be the number of successes in performing a sequence of independent Bernoulli trials where is the probability of success. In other words, has a binomial distribution with parameters and .
If the random experiment requires more than failures to reach the rth success, there are at most successes in the first trails. Thus the survival function of is the same as the cdf of a binomial distribution. Equivalently, the cdf of is the same as the survival function of a binomial distribution. We have the following:
The relation is analogous to the relationship between the Gamma distribution and the Poisson distribution. Recall that a Gamma distribution with shape parameter and scale parameter , where is a positive integer, can be interpreted as the waiting time until the nth change in a Poisson process. Thus, if the nth change takes place after time , there can be at most arrivals in the time interval . Thus the survival function of this Gamma distribution is the same as the cdf of a Poisson distribution. The relation is analogous to the following relation.
A previous post on the negative binomial distribution is found here.