“negative binomial distribution”的概念、定义、翻译、参考文献-科学参考

negative binomial distribution

Physics

See Pascal’s distribution.

Mathematics

The generalization of the geometric distribution to the waiting time for the kth success in a sequence of independent experiments, all with the same probabilities of success. The probability mass function is given by

$Pr {X = r} = (\begin{matrix} r - 1 \\ k - 1 \end{matrix}) p^{k} {(1 - p)}^{r - k} .$

This is because the kth success will be on the rth trial if the rth trial is a success and there are k–1 successful trials in the previous r–1 trials. See appendix 15.

Statistics

For trials classified as ‘success’ or ‘failure’, the distribution of X, the number of trials required in order to obtain n successes. The trials are presumed to be independent and it is assumed that each trial has the same probability of success, p (≠ 0 or 1). The probability function is $negative binomial distribution$ The mean of this distribution is n/p and the variance is n(1−p)/p².
Special cases of the distribution were discussed by Pascal in 1679. The name ‘negative binomial’ arises because the probabilities are successive terms in the binomial expansion of (P−Q)⁻ⁿ, where P=1/p and Q=(1− p)/p. Writing Y=X−n, an equivalent form for the distribution is $negative binomial distribution$ The variable X may be regarded as the sum of n independent geometric variables, each with parameter p. The case n=1 therefore corresponds to the geometric distribution.
The arc-sinh transformation, suggested by Anscombe in 1948, $negative binomial distribution$ results in a random variable S having an approximate standard normal distribution.
Negative binomial distribution. In each case n=5; the shape is dependent upon the value of p.

单词	negative binomial distribution
释义	negative binomial distribution Physics See Pascal’s distribution. Mathematics The generalization of the geometric distribution to the waiting time for the kth success in a sequence of independent experiments, all with the same probabilities of success. The probability mass function is given by $Pr {X = r} = (\begin{matrix} r - 1 \\ k - 1 \end{matrix}) p^{k} {(1 - p)}^{r - k} .$ This is because the kth success will be on the rth trial if the rth trial is a success and there are k–1 successful trials in the previous r–1 trials. See appendix 15. Statistics For trials classified as ‘success’ or ‘failure’, the distribution of X, the number of trials required in order to obtain n successes. The trials are presumed to be independent and it is assumed that each trial has the same probability of success, p (≠ 0 or 1). The probability function is $negative binomial distribution$ The mean of this distribution is n/p and the variance is n(1−p)/p². Special cases of the distribution were discussed by Pascal in 1679. The name ‘negative binomial’ arises because the probabilities are successive terms in the binomial expansion of (P−Q)⁻ⁿ, where P=1/p and Q=(1− p)/p. Writing Y=X−n, an equivalent form for the distribution is $negative binomial distribution$ The variable X may be regarded as the sum of n independent geometric variables, each with parameter p. The case n=1 therefore corresponds to the geometric distribution. The arc-sinh transformation, suggested by Anscombe in 1948, $negative binomial distribution$ results in a random variable S having an approximate standard normal distribution. Negative binomial distribution. In each case n=5; the shape is dependent upon the value of p.
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