What is the collection of parameters for the negative binomial distribution when summing independent geometric distributions with parameter p?

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The correct choice is based on the relationship between the negative binomial distribution and the geometric distribution. The negative binomial distribution describes the number of trials needed to achieve a fixed number of successes (denoted as 'r') in a sequence of Bernoulli trials, wherein each trial has the same probability of success (denoted as 'p').

In the context of summing independent geometric distributions, each geometric distribution can be viewed as describing the number of failures before the first success in a series of Bernoulli trials. When you add independent geometrically distributed random variables, you essentially model the total number of trials until the r-th success occurs, which is exactly what the negative binomial distribution does.

Here, 'r' represents the number of successes you want to achieve, and 'p' is the probability of success on each trial. The choice stating 'r = n and p' aligns with this definition, where 'n' is the number of successes you are summing from the independent geometric distributions, and 'p' is the consistent probability of success in those trials.

Thus, the parameters for the negative binomial distribution that results from summing these independent geometric distributions are appropriately represented as 'r' equal to the total number of successes