How Do You Spell NEGATIVE BINOMIAL DISTRIBUTIONS?

Pronunciation: [nˈɛɡətˌɪv ba͡ɪnˈə͡ʊmɪəl dˌɪstɹɪbjˈuːʃənz] (IPA)

Negative binomial distributions (nɛɡətɪv baɪnoʊmiəl dɪstrɪbjuʃənz) are probability distributions that characterize the number of failures in a fixed number of independent trials before a specified number of successes are achieved. The term "negative" refers to the fact that the distribution counts the number of failures before reaching the desired number of successes. The term "binomial" refers to the underlying distribution that models the probability of success or failure in each trial. The spelling of the word is derived from its mathematical origins and is pronounced according to the International Phonetic Alphabet.

NEGATIVE BINOMIAL DISTRIBUTIONS Meaning and Definition

1. A negative binomial distribution is a probability distribution that describes the number of successful trials required until a specified number of failures is observed. It is used to model situations where the events are repeated until a certain number of failures occur.

   In a negative binomial distribution, the probability of success in each trial remains constant. The distribution is characterized by two parameters: the number of failures (r) and the probability of success (p). The number of failures represents the specific number of failures that need to be observed before the experiment stops. The probability of success represents the likelihood of a successful event on any given trial.

   The negative binomial distribution is similar to the binomial distribution, but differs in that the binomial distribution focuses on the number of successes in a fixed number of trials, while the negative binomial distribution focuses on the number of trials required until a fixed number of failures is reached.

   The probability mass function of a negative binomial distribution gives the probability of observing a specific number of successes (k) before the rth failure is achieved. It can be calculated using a formula that involves the binomial coefficient, the probability of success, and the probability of failure.

   Negative binomial distributions are commonly used in areas such as quality control, economics, and biology to model events that follow a sequence of successes and failures until a desired outcome is reached.

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