Negative Binomial Mass Function

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A Negative Binomial Mass Function is a discrete probability function where p is the probability of a success in a sequence independent Bernoulli trials before r failures.



  • (Wikipedia, 2011) ⇒
    • QUOTE: In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified (non-random) number [math]\displaystyle{ r }[/math] of failures occurs. For example, if one throws a die repeatedly until the third time “1” appears, then the probability distribution of the number of non-“1”s that had appeared will be negative binomial. The Pascal distribution (after Blaise Pascal) and Polya distribution (for George Pólya) are special cases of the negative binomial. There is a convention among engineers, climatologists, and others to reserve “negative binomial” in a strict sense or “Pascal” for the case of an integer-valued stopping-time parameter [math]\displaystyle{ r }[/math], and use “Polya” for the real-valued case. The Polya distribution more accurately models occurrences of “contagious” discrete events, like tornado outbreaks, than the Poisson distribution.
    • QUOTE: Suppose there is a sequence of independent Bernoulli trials, each trial having two potential outcomes called “success” and “failure”. In each trial the probability of success is [math]\displaystyle{ p }[/math] and of failure is (1 − p). We are observing this sequence until a predefined number [math]\displaystyle{ r }[/math] of failures has occurred. Then the random number of successes we have seen, [math]\displaystyle{ X }[/math], will have the negative binomial (or Pascal) distribution: [math]\displaystyle{ X\ \sim\ \text{NB}(r,\, p) }[/math]