Negative Binomial Distribution

The negative binomial distribution models the number of trials needed to achieve r successes, assuming each trial has the same probability of success, θ. Here, successes are fixed and trials are variable. If you repeat independent trials until the rᵗʰ success occurs, the probability that this happens after x failures (i.e., on the x + rth trial) is given by: 
 

\(f(x) = \left( \frac{x + r - 1}{r - 1} \right) p r^x, \quad x = 0, 1, 2, \dots \)

Where:
x = number of failures before the r-th success
r = total number of successes we want to observe
p = probability of success on a single trial
q = probability of failure on a single trial = 1 − p
 

The objective of the negative binomial distribution is to help us figure out how many trials it might take to get a certain number of successes when each trial has the same chance of success. It depends mainly on two things: 

• r - the required number of successes, which should be a whole number.
• p - the probability of success in each trial (between 0 and 1).
• These two values shape how the distribution looks and how it behaves.

For example:
Let’s say a basketball player wants to make 4 successful free throws. 

• So, r = 4 
• Let’s say their chance of making a shot is p = 0.3 (30%)

This setup tells us how many total attempts we might expect before the player gets those 4 makes. 

• If we increase r, the player will need more tries, and the results can vary more. 
• If we increase p, the player will probably need fewer attempts, and the number of trials becomes more predictable. 

The properties of the negative binomial distribution are as follows:

• The experiment consists of independent and identical trials, repeated until a fixed number of successes (r) is observed. 

• Each trial results in either a success or a failure. 

• The probability of success (p) stays the same in every trial. The probability of failure is q = 1 - p. 

• The outcome of one trial does not influence the outcome of another. 

• The experiment continues until a specific number of successes (r) is reached. The value of r is chosen in advance. 

• The total number of trials varies and is equal to x + r, where x is the number of failures before the final (r-th) success.

The probability density function of the negative binomial distribution gives the probability of observing a specific number of failures before reaching a fixed number of successes. It is given by: 

\(P(X = x) = \left( \frac{x + r - 1}{r - 1} \right) p^r (1 - p)^x, \quad x = 0, 1, 2, \dots \)

Where,
x = number of failures before rth success
r = required number of successes
p = probability of success in each trial 
(1-p) = q = probability of failure, and 
The binomial coefficient shows how many different ways the failures and successes can be ordered before the final success occurs. 
It describes the probability that the rth success occurs exactly on trial x + r.
 

The mean and variance tell us what to expect from the number of trials needed to reach a set number of successes.

• Mean: 
  Mean = rp. It represents the average number of trials needed to get r successes, where p is the probability of success in each trial. 

• Variance:
  Variance = r(1 - p)p2. It shows how much the number of trials might vary around the average.

Example:
If a player has a 40% chance of scoring on each attempt and wants to score 3 times, then:

• Mean = 3/0.4 = 7.5 trials on average (p = 0.4 and r = 3)
• Variance = 3(1 - 0.4) / (0.4)2 = 3(0.6) / 0.16 = 11.25 
   

Negative binomial distribution has a wide range of applications in various fields like hospitality, R&D, and marketing. Here, we’ve mentioned a few applications.

• Customer service/ sales calls: A salesperson may want to know how many phone calls they need to make before successfully closing 5 deals, assuming each call has a fixed chance of success.

• Determining the number of patients in medical trials: In clinical research, the distribution can model how many patients need to be treated until a certain number of successful outcomes (e.g., patients cured) are observed, with each treatment having a fixed success probability.

• Quality Control in Manufacturing: Factories can use this method to estimate the number of items that need to be inspected before finding a certain number of defective pieces, especially if the defects occur randomly.

• Student testing in educational institutions: A teacher can find out how many attempts a student will need to pass a certain number of modules, given a certain chance of passing each one.

• Key performance indicators in digital marketing:Negative binomial distribution can help predict how much website traffic or ad clicks are required before a set number of purchases occur.