14.1 Properties of the Negative Binomial Distribution

The Negative Binomial Distribution is a fundamental statistical tool used to model the number of trials required to achieve a fixed number of successes in a sequence of independent Bernoulli trials. It is particularly useful in scenarios where the number of attempts is the primary focus, rather than a fixed number of trials.

Core Characteristics

The Negative Binomial Distribution is defined by the following key properties:

1. Fixed Number of Successes ($r$): The distribution's primary focus is on the total number of trials needed to achieve a specific, predetermined number of successes, denoted by '$r$'. The experiment continues until this exact count of successes is reached.

2. Two Possible Outcomes per Trial: Each individual trial within the experiment has only two mutually exclusive outcomes:

   • Success: The desired outcome occurs.
   • Failure: The desired outcome does not occur.

3. Constant Probability of Success ($p$): The probability of achieving a success in any given trial remains constant throughout the entire sequence of trials. This stable probability, denoted by '$p$', is crucial for the distribution's validity.

4. Consistent Probability of Failure ($q$): Similarly, the probability of failure in any trial is also constant and is directly related to the probability of success. The probability of failure, denoted by '$q$', is calculated as: $$q = 1 - p$$ This relationship ensures that the probabilities of success and failure are complementary and sum to 1.

5. Independence of Trials: A critical assumption of the Negative Binomial Distribution is that each trial is independent of all other trials. This means that the outcome of one trial has absolutely no influence on the outcome of any subsequent trial. This independence is vital for accurate probability calculations.

6. Variable Number of Trials ($X$): Unlike the Binomial Distribution where the number of trials is fixed beforehand, the total number of trials in a Negative Binomial scenario is a random variable. The process continues until the '$r$'th success is observed, meaning the total number of trials can vary from one instance of the experiment to another.

7. Relationship Between Failures and Total Trials: If '$x$' represents the number of failures that occur before the '$r$'th success is achieved, then the total number of trials ($n$) required to reach the '$r$'th success is given by: $$n = x + r$$ In this formulation, the last trial is always a success.

Example Scenario: Quality Control

Imagine a factory producing light bulbs, and the desired quality standard is that 10 bulbs must pass a specific test (i.e., $r=10$ successes). The probability that a single bulb passes the test is $p = 0.8$. The probability that a bulb fails is $q = 1 - 0.8 = 0.2$.

The factory continues testing bulbs until they have 10 bulbs that pass. The number of bulbs tested after the 10th passing bulb is not considered. The number of bulbs tested could vary. For instance:

• If the first 10 bulbs all pass, the total number of trials is 10 ($x=0$ failures).
• If the 11th bulb tested is the 10th success, then there were 10 successes and 1 failure, making the total number of trials 11 ($x=1$ failure).
• If the 15th bulb tested is the 10th success, then there were 10 successes and 5 failures, making the total number of trials 15 ($x=5$ failures).

Applications

The properties of the Negative Binomial Distribution make it an ideal model for scenarios where one is interested in modeling the number of attempts or trials required to achieve a fixed number of successful outcomes. Common applications include:

• Quality Control: Determining how many items need to be inspected to find a certain number of defective items.
• Clinical Trials: Modeling the number of patients required to observe a specific number of treatment successes.
• Customer Acquisition Campaigns: Estimating the number of leads or interactions needed to secure a fixed number of new customers.
• Reliability Engineering: Analyzing the time until a system experiences a certain number of failures.
• Gambling and Games: Modeling the number of bets or rounds needed to achieve a specific number of wins.

Conclusion

The Negative Binomial Distribution is a powerful and versatile tool for modeling count data where the stopping condition is the achievement of a fixed number of successes. Its distinct characteristics, particularly the variable number of trials and the constant probabilities of success and failure, allow it to accurately represent many real-world processes where outcomes are observed sequentially until a specific target is met.