What is the expected value of a negative binomial distribution?

What is the expected value of a negative binomial distribution?

• A. E[X] = rq/p
• B. E[X] = p/q
• C. E[X] = r + x
• D. E[X] = rq

Answer: (A)

In a negative binomial distribution, the expected value represents the expected number of trials required to achieve a fixed number of successes, where each trial has a distinct probability of success. The parameters of the negative binomial distribution include \( r \), which denotes the number of successes, \( p \), the probability of success in each trial, and \( q \) which represents the probability of failure (where \( q = 1 - p \)). The formula for calculating the expected value \( E[X] \) in a negative binomial distribution is derived from the structure of the distribution. Specifically, it combines the number of successes \( r \) with the average number of trials needed for each success, which is quantified as \( rq/p \). This expression represents that for every success, the expected number of trials is influenced by the likelihood of failing in trials until achieving the required \( r \) successful outcomes. Thus, when \( r \) success outcomes are pursued, each of these successes on average requires \( q/p \) trials (since \( q \) is the probability of failure and \( p \) is the probability of success). Hence, the aggregate expected number of trials for all \( r \) successes accumulates to \(

In a negative binomial distribution, the expected value represents the expected number of trials required to achieve a fixed number of successes, where each trial has a distinct probability of success. The parameters of the negative binomial distribution include ( r ), which denotes the number of successes, ( p ), the probability of success in each trial, and ( q ) which represents the probability of failure (where ( q = 1 - p )).

The formula for calculating the expected value ( E[X] ) in a negative binomial distribution is derived from the structure of the distribution. Specifically, it combines the number of successes ( r ) with the average number of trials needed for each success, which is quantified as ( rq/p ). This expression represents that for every success, the expected number of trials is influenced by the likelihood of failing in trials until achieving the required ( r ) successful outcomes.

Thus, when ( r ) success outcomes are pursued, each of these successes on average requires ( q/p ) trials (since ( q ) is the probability of failure and ( p ) is the probability of success). Hence, the aggregate expected number of trials for all ( r ) successes accumulates to (