The negative binomial distribution is a probability distribution that models the number of independent and identical Bernoulli trials required to obtain a fixed number of successes, denoted by the parameter "r." In simpler terms, it describes the number of failures that occur before a specific number of successes is achieved.
In this distribution, each trial results in either success or failure, with success having a fixed probability denoted by "p." The negative binomial distribution differs from the binomial distribution in that the latter focuses on the number of trials needed to achieve a specific number of successes, while the former focuses on the number of failures before that specific number is reached.
The probability mass function of the negative binomial distribution is given by P(X = k) = (k + r - 1)C(r - 1) p^r (1-p)^k, where k is the number of failures, r is the desired number of successes, p is the probability of success, and C is the combination function.
The mean (μ) of the negative binomial distribution is given by μ = r(1-p)/p, while the variance (σ²) is given by σ² = r(1-p)/p². These parameters provide important insights into the shape, spread, and location of the distribution.
This distribution finds applications in various fields, such as quality control, reliability analysis, and queueing theory. It allows researchers and statisticians to analyze and predict the number of trials needed to achieve a certain level of success, providing valuable insights for decision-making and planning purposes.
