What does the additive property of the negative binomial distribution state?

What does the additive property of the negative binomial distribution state?

• A. The sum follows a negative binomial distribution with parameters as the sum of the ri and p.
• B. The sum is uniformly distributed.
• C. The sum follows a Poisson distribution.
• D. The parameters remain constant regardless of independence.

Answer: (A)

The additive property of the negative binomial distribution states that if you have independent negative binomial random variables, the sum of these variables will also follow a negative binomial distribution. Specifically, if you have two independent negative binomial distributions with parameters (r1, p) and (r2, p), the sum of these two distributions will yield a new negative binomial distribution with parameters (r1 + r2, p). This reflects that the total number of successes necessary to achieve a specified number of failures remains consistent, and the probability of success (p) remains the same across the distributions involved. This property highlights the nature of the negative binomial distribution, which counts the number of trials needed to achieve a fixed number of successes in a sequence of independent and identically distributed Bernoulli trials. Therefore, as the parameters of the individual distributions are summed together, they define a new negative binomial distribution, confirming the additive property accurately stated in the correct answer.

The additive property of the negative binomial distribution states that if you have independent negative binomial random variables, the sum of these variables will also follow a negative binomial distribution. Specifically, if you have two independent negative binomial distributions with parameters (r1, p) and (r2, p), the sum of these two distributions will yield a new negative binomial distribution with parameters (r1 + r2, p). This reflects that the total number of successes necessary to achieve a specified number of failures remains consistent, and the probability of success (p) remains the same across the distributions involved.

This property highlights the nature of the negative binomial distribution, which counts the number of trials needed to achieve a fixed number of successes in a sequence of independent and identically distributed Bernoulli trials. Therefore, as the parameters of the individual distributions are summed together, they define a new negative binomial distribution, confirming the additive property accurately stated in the correct answer.