Final answer:
A negative binomial random variable X with parameters r and p represents the number of trials needed to achieve r successes, with p being the success probability of each independent trial. This differs from the binomial distribution which focuses on the number of successes in a fixed number of trials.
Step-by-step explanation:
Let X1, X2, ..., Xn be a random sample of a negative binomial random variable X with parameters r and p. The random variable X represents the number of trials needed to achieve a certain number of successes, r, in a sequence of Bernoulli trials. The parameter p denotes the probability of a success on a single trial.
The negative binomial distribution is a generalization of the geometric distribution and applies to experiments where we are interested in the number of trials required to achieve the r-th success, given a constant success probability p for each independent trial. This contrasts with the binomial distribution, where X ~ B(n, p) represents the number of successes in a fixed number n of independent trials, with each trial having a success probability p.
It's important to understand that the negative binomial distribution captures the variability in the number of trials until achieving the desired number of successes, rather than simply the count of successes in a fixed number of trials, which is the focus of the binomial distribution.