What kind of distribution results when summing n independent geometric distributions with parameter p?

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When summing ( n ) independent geometric distributions with parameter ( p ), the resulting distribution is a negative binomial distribution. Each geometric distribution represents the number of trials until the first success, which is a specific case of the negative binomial distribution. The negative binomial distribution describes the number of trials needed to achieve a fixed number of successes (in this case, ( n ) successes).

In a negative binomial distribution, the parameters usually consist of ( r ), the number of successes, and ( p ), the probability of success on each trial. If we are summing ( n ) independent geometric random variables, we are essentially looking for the number of trials required to achieve ( n ) successes, leading us directly to the negative binomial framework.

In contrast, the other distributions mentioned do not fit this scenario. The binomial distribution deals with a fixed number of trials and counts the number of successes within those trials, not the trials needed to achieve them. The normal distribution pertains to large sample sizes approximating distributions of sums or averages under certain conditions, but it doesn’t apply directly in this context of summing geometric variables. Lastly, the Poisson distribution is used for modeling the number of events occurring in a fixed