The negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent Bernoulli trials before a specified number of failures occur. It is a generalization of the geometric distribution, which models the number of trials until the first success occurs.
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The negative binomial distribution is characterized by two parameters: the number of failures (r) and the probability of success in each trial (p).
The negative binomial distribution is often used to model the number of failures before a certain number of successes are achieved in a series of independent Bernoulli trials.
The negative binomial distribution is a generalization of the geometric distribution, where the geometric distribution is a special case of the negative binomial distribution with r = 1.
The probability mass function (PMF) of the negative binomial distribution is given by the formula: $P(X = x) = \binom{x - 1}{r - 1} p^r (1 - p)^{x - r}$, where x is the number of trials, r is the number of failures, and p is the probability of success in each trial.
The negative binomial distribution has applications in various fields, such as biology, finance, and queuing theory, where the number of failures before a certain number of successes is of interest.
Review Questions
Explain the relationship between the negative binomial distribution and the geometric distribution.
The negative binomial distribution is a generalization of the geometric distribution. While the geometric distribution models the number of trials until the first success occurs, the negative binomial distribution models the number of trials until a specified number of failures (r) occur. The negative binomial distribution reduces to the geometric distribution when the number of failures (r) is set to 1.
Describe the parameters of the negative binomial distribution and how they affect the distribution.
The negative binomial distribution is characterized by two parameters: the number of failures (r) and the probability of success in each trial (p). The number of failures (r) determines the shape of the distribution, with higher values of r resulting in a more skewed distribution. The probability of success (p) affects the overall scale of the distribution, with higher values of p leading to a lower number of trials on average before the specified number of failures occurs.
Explain how the negative binomial distribution can be used to model real-world scenarios and provide an example application.
The negative binomial distribution can be used to model situations where the number of failures before a certain number of successes is of interest. For example, in quality control, the negative binomial distribution can be used to model the number of defective items that need to be inspected before a certain number of acceptable items are found. Another application is in insurance, where the negative binomial distribution can be used to model the number of claims made by policyholders before a certain number of claims are denied.
A Bernoulli trial is a random experiment with two possible outcomes, typically labeled as 'success' and 'failure,' where the probability of success remains constant across trials.
The geometric distribution is a discrete probability distribution that models the number of trials until the first success occurs in a sequence of independent Bernoulli trials.
Probability Mass Function (PMF): The probability mass function (PMF) of a discrete random variable describes the probability of each possible value the variable can take.