Negative Binomial Example

A manufacturer of jet engine turbines has an order to produce 50 turbines. Since about 20% of the turbines do not make it past the high-velocity spin test, the manufacturer will actually have to produce more than 50 turbines.

Matching these conditions with the negative binomial distribution:

The number of turbines to produce (trials) is not fixed.
The manufacturer will continue to produce turbines until the 50th one has passed the spin test.
The probability of success (80%) is the same for each test.

These conditions match those of the negative binomial distribution.

The negative binomial distribution has two parameters: Probability and Shape. The Shape parameter specifies the r th successful occurrence. In this example you would enter 0.8 for the Probability parameter (80% success rate of the spin test) and 50 for the Shape parameter (Figure 104, Negative Binomial Distribution).

Figure 104. Negative Binomial Distribution

This figure displays a negative binomial distribution.

Some characteristics of the negative binomial distribution:

When Shape = 1, the negative binomial distribution becomes the geometric distribution.
The sum of any two negative binomial distributed variables is a negative binomial variable.
Another form of the negative binomial distribution, sometimes found in textbooks, considers only the total number of failures until the r th successful occurrence, not the total number of trials. To model this form of the distribution, subtract out r (the value of the shape parameter) from the assumption value using a formula in the worksheet.