Formula of Calculating Mean for Negative Binomial Probability Distribution Is

The negative binomial distribution is a probability distribution that models the number of trials needed to achieve a given number of successes. The mean (expected value) of this distribution provides insight into the average number of trials required to achieve the desired number of successes.

What is the Negative Binomial Distribution?

The negative binomial distribution is a discrete probability distribution that describes the number of trials needed to achieve a specified number of successes in repeated, independent Bernoulli trials. It is often used in scenarios where:

You need a certain number of successes before stopping
Each trial has the same probability of success
Trials are independent

This distribution is different from the binomial distribution, which models the number of successes in a fixed number of trials.

Mean Formula

The mean (expected value) of a negative binomial distribution is calculated using the following formula:

Mean = r / p

Where:

r = number of successes needed
p = probability of success on an individual trial

This formula shows that the mean number of trials needed to achieve r successes is simply the number of successes divided by the probability of success in each trial.

How to Calculate the Mean

To calculate the mean of a negative binomial distribution:

Identify the number of successes needed (r)
Determine the probability of success in each trial (p)
Divide the number of successes by the probability of success (r / p)

Important: The probability p must be between 0 and 1 (0 < p < 1), and r must be a positive integer (r ≥ 1).

Worked Example

Suppose you're testing a new website feature and want to know how many visitors you need to test to achieve 5 successful conversions, with each visitor having a 20% chance of converting. Calculate the expected number of visitors needed.

Using the formula:

Mean = r / p = 5 / 0.20 = 25

This means you would expect to need to test approximately 25 visitors to achieve 5 successful conversions.

FAQ

What is the difference between binomial and negative binomial distributions?

The binomial distribution models the number of successes in a fixed number of trials, while the negative binomial distribution models the number of trials needed to achieve a fixed number of successes.

When should I use the negative binomial distribution?

Use the negative binomial distribution when you need to model scenarios where you're counting the number of trials until a certain number of successes occur, such as in quality control, medical testing, or A/B testing.

What happens if the probability of success is very low?

If the probability of success is very low, the mean number of trials needed will be very high, as you'll need many trials to achieve even a small number of successes.