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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Negative binomial distribution ： ウィキペディア英語版 Negative binomial distribution In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted ''r'') occurs. For example, if we define a "1" as failure, all non-"1"s as successes, and we throw a dice repeatedly until the third time “1” appears (''r'' = three failures), then the probability distribution of the number of non-“1”s that had appeared will be a 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 ''r'', and use “Polya” for the real-valued case. For occurrences of “contagious” discrete events, like tornado outbreaks, the Polya distributions can be used to give more accurate models than the Poisson distribution by allowing the mean and variance to be different, unlike the Poisson. “Contagious” events have positively correlated occurrences causing a larger variance than if the occurrences were independent, due to a positive covariance term. ==Definition== 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 ''p'' and of failure is (1 − ''p''). We are observing this sequence until a predefined number ''r'' of failures has occurred. Then the random number of successes we have seen, ''X'', will have the negative binomial (or Pascal) distribution: : $$ X\sim\operatorname(r; p) When applied to real-world problems, outcomes of ''success'' and ''failure'' may or may not be outcomes we ordinarily view as good and bad, respectively. Suppose we used the negative binomial distribution to model the number of days a certain machine works before it breaks down. In this case “success” would be the result on a day when the machine worked properly, whereas a breakdown would be a “failure”. If we used the negative binomial distribution to model the number of goal attempts a sportsman makes before scoring a goal, though, then each unsuccessful attempt would be a “success”, and scoring a goal would be “failure”. If we are tossing a coin, then the negative binomial distribution can give the number of heads (“success”) we are likely to encounter before we encounter a certain number of tails (“failure”). In the probability mass function below, p is the probability of success, and (1-p) is the probability of failure. The probability mass function of the negative binomial distribution is : $$ f(k; r, p) \equiv \Pr(X = k) = \binom p^k(1-p)^r \quad\textk = 0, 1, 2, \dotsc Here the quantity in parentheses is the binomial coefficient, and is equal to : $$ \binom = \frac = \frac. This quantity can alternatively be written in the following manner, explaining the name “negative binomial”: : $$ \frac = (-1)^k \frac = (-1)^k\binom. \qquad ( *) To understand the above definition of the probability mass function, note that the probability for every specific sequence of ''k'' successes and ''r'' failures is , because the outcomes of the ''k'' + ''r'' trials are supposed to happen independently. Since the ''r''th failure comes last, it remains to choose the ''k'' trials with successes out of the remaining ''k'' + ''r'' − 1 trials. The above binomial coefficient, due to its combinatorial interpretation, gives precisely the number of all these sequences of length ''k'' + ''r'' − 1. The following recurrence relation holds: : $\backslash left\backslash$ (k+1) \Pr (k+1)-p \Pr (k) (k+r)=0, \\() \Pr (0)=(1-p)^r \end\right\} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Negative binomial distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.