Negative Binomial Distribution : Properties, Applications and Examples

What is Negative Binomial Distribution?

The Negative Binomial Distribution deals with the number of trials needed to achieve a specific number of successes. It helps determine how many trials it takes to reach a desired number of successes. In this distribution, we define the number of trials and the probability of success in each trial. In simple terms, it tells us how many attempts are required to achieve a set number of successes, considering the probability of success in each attempt. Negative Binomial Distribution is denoted by (k,).

It is similar to a binomial distribution but with one key difference, in a binomial distribution, the number of trials is fixed, while in the negative binomial distribution, the number of successes is fixed.

Properties of Negative Binomial Distribution

The negative binomial distribution has specific characteristics,

• It involves a total of ‘n’ trials.
• Each trial has two possible outcomes, success and failure.
• The probability of success (denoted as ‘p’) is the same for each trial.
• The probability of failure (denoted as ‘q’) is also consistent across trials, with p + q equaling 1.
• Trials are independent; the outcome of one trial doesn’t influence others.
• The experiment continues until a predetermined number of ‘r’ successes are achieved.
• In total, there are ‘x + r’ repeated trials to reach the desired number of successes.

Probability Density Function (PDF) of Negative Binomial Distribution

The Negative Binomial Distribution‘s Probability Density Function (PDF) describes the likelihood of getting a certain number of successes before a specific number of failures happen in a series of independent trials. The formula for the PDF looks like this,

where, X is the number of trials until k^th success and is the probability of success in a single trial.

This formula helps us figure out the probability of observing a specific number of successes (k) before encountering r failures in a sequence of trials, where is the probability of success in each trial.

Mean and Variance of Negative Binomial Distribution

The mean and variance of the Negative Binomial Distribution can help us understand the average and spread of the number of trials needed to achieve a certain number of successes.

I. Mean of Negative Binomial Distribution

The mean or average (μ) of the Negative Binomial Distribution is given by dividing the number of successes needed (k) by the probability of success (). It tells us, on average, how many trials we can expect to have before getting r success.

II. Variance of Negative Binomial Distribution

Variance measures how much the actual number of trials required might deviate from the average. The formula involves multiplying the number of successes needed (k) by the probability of failure (1−) and then dividing by the square of the probability of success (). A higher variance indicates more variability in the number of trials needed.

Applications of Negative Binomial Distribution

The Negative Binomial Distribution finds applications in various aspects of business statistics. Here are a few practical scenarios where it is commonly used,

1. Project Management: In project management, the Negative Binomial Distribution can be applied to estimate the number of trials (attempts or tasks) required to complete a project with a certain number of successful outcomes (milestones achieved or tasks completed). This helps in project planning and resource allocation.

2. Quality Control: In manufacturing and quality control, the distribution can be used to model the number of defective items produced before reaching a certain number of acceptable items. This aids in setting quality standards and optimizing production processes.

3. Customer Service and Call Centers: The Negative Binomial Distribution is often employed to model the number of customer service calls a representative needs to handle before resolving a certain number of issues. This is valuable for workforce management and optimizing service efficiency.

4. Marketing Campaigns: Marketers can use the Negative Binomial Distribution to predict the number of attempts (such as advertisement exposures or promotional events) needed to achieve a specific number of desired responses, like customer purchases or sign-ups.

5. Insurance and Risk Management: In the insurance industry, the distribution can be utilized to model the number of claims or losses a company might experience before reaching a certain level of profitability. This aids in risk assessment and setting appropriate insurance premiums.

6. Inventory Management: Businesses dealing with inventory can use the Negative Binomial Distribution to model the number of orders or deliveries needed to restock inventory before selling a certain quantity of products. This helps in maintaining optimal stock levels and minimizing holding costs.

Examples of Negative Binomial Distribution

Example 1:

The probability of Mary solving a puzzle correctly is 75%. What is the probability that Mary solves the puzzle correctly for the fifth time in the first eight attempts?

Solution:

Here, the occurrence of success is at the 5^th time, so k = 5.

Number of Trials, X = 8.

The probability of getting a success = 0.75 and the probability of getting a failure will be 1 – = 0.25

So, 

P (X = 8) = 35 × 0.2373046875 × 0.015625

P (X = 8) = 0.12977600097

P (X = 8) = 0.13

∴ the probability that Mary solves the puzzle correctly for the fifth time in the first eight attempts is approximately 0.13.

Example 2:

The probability of Alex hitting the target in archery is 90%. What is the probability that Alex hits the target for the second time in four attempts?

Solution:

Here, the occurrence of hitting the target is at 2^nd time, so k = 2.

Number of Trials, X = 4.

The probability of getting a success = 0.90 and the probability of getting a failure will be 1 – = 0.10

So, 

P (X = 4) = 3 x 0.81 x 0.01

P (X = 4) = 0.0243

∴ the probability that Alex hits the target for the second time in the first four attempts is 0.02.