The Negative Binomial Distribution

• The Negative Binomial Distribution (NBD) is a discrete probability distribution which models the number of successes before a specified number of failures occur in Bernoulli trials.
• The Bernoulli trials are assumed to be independent; the outcome of one trial does not affect the outcome of another.
• The distribution is defined by two parameters: the number of failures (r) we wish to observe, and the probability of success (p) on an individual trial.
• Probability mass function of NBD is P(X = x) = C(x-1, r-1) * p^r * (1-p)^(x-r) where x is the total number of trials, r is the number of failures, p is the probability of success, and C() indicates a combination.
• Mean (expected value) of NBD is E(X) = r/p where p is the probability of success and r is the number of failures.
• Variance of NBD is V(X) = r(1 - p) / p^2 where p is the probability of success and r is the number of failures.
• The NBD is skewed to right. It is positively skewed when p>1/2 and negatively skewed when p<1/2.
• The Negative Binomial distribution is used in scenarios where we are conducting trials until a predefined number of failures occur, making it useful for reliability testing or quality control.
• It is a more general case of the Geometric Distribution, which is a special case of the NBD where r=1.
• It’s important to distinguish between Negative Binomial Distribution and Binomial Distribution - they are often confused due to similar names but they model different situations and possess different formulae.
• Real-life examples of NBD scenarios include the number of times one needs to roll a dice until a certain number of sixes (failures) are observed, or the number of sales calls it takes until a certain number of rejections (failures) are received.
• As with any probability distribution, in-depth understanding of the Negative Binomial distribution will allow correct and complex application to a variety of real-world problems and scenarios.