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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ 第101行：
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第161行： − 在概率论和统计学中，参数 n 和 p 的二项分布是 n 个独立实验序列中成功次数的离散概率分布，每个实验询问一个 yes-no 问题，每个实验都有自己的布尔值结果: success/yes/true/one (带概率 p)或 failure/no/false/zero (带概率 q = 1-p)。+ + 第167行：
第168行： − 一个单一的成功/失败的实验也被称为伯努利实验或伯努利实验，一系列的结果被称为伯努利过程; 对于一个单一的实验，例如，n = 1，二项分布是一个伯努利分布。二项分布是流行的统计显著性二项检验的基础。+ − 第175行：
第175行： − 二项分布经常被用来模拟 n 大小的样本中的成功数量，这些样本是用 n 大小的种群中的替代物抽取的。如果抽样没有更换，抽样就不是独立的，所以得到的分布是超几何分布，而不是二项式。然而，对于 n 比 n 大得多的情况，二项分布仍然是一个很好的近似值，并且被广泛使用。+ − 第191行：
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第222行： − 是二项式系数，因而得名。这个公式可以理解为。K 的成功发生在概率 pk 和 n-k 的失败发生在概率(1-p) n-k 的情况下。然而，k 的成功可以发生在 n 个试验中的任何地方，并且在 n 个试验序列中有不同的 k 的成功分配方法。+ 第231行：
第230行： − 在为二项分布概率创建参考表时，通常表中最多填充 n/2的值。这是因为对于 k > n/2，概率可以通过它的补来计算+ 第259行：
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第330行： − 它也可以用正规化的不完全 beta 函数来表示，如下:+ 第387行：
第386行： − 下面给出了累积分布函数的一些封闭形式界。+ 第401行：
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第416行： − 这是从线性的期望值，以及事实上是相同的伯努利随机变量的和，每个与期望值。换句话说，如果 x1，ldots，xn 是带参数的伯努利随机变量，那么 x = x1 + cdots + xn+ 第445行：
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第454行： − 前6个中心时刻由+ 第515行：
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第691行： − 从渐近的角度来看，这个界限是相当紧密的; 参见+ 第715行：
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第1,213行： − 给定一个统一的先验，给定具有观察到的成功事件的独立事件的成功概率的后验概率是一个 Β分布。+ 第1,223行：
第1,225行： − 随机数产生的方法，其中的边缘分布是一个二项分布。+ 第1,231行：
第1,233行： − 从二项分布中产生随机样本的一种方法是使用反演算法。要做到这一点，我们必须计算从到的所有值的概率。(为了包含整个样本空间，这些概率之和应该接近于一。)然后，通过使用伪随机数生成器来生成0到1之间的样本，人们可以使用在第一步计算出的概率将计算出的样本转换成离散数。+ 第1,262行：
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无编辑摘要
此词条由南风翻译
{{short description|Probability distribution}}
{{short description|Probability distribution}}
| variance = npq
| variance = npq
| variance = npq
| 方差 = npq
| skewness = <math>\frac{q-p}{\sqrt{npq}}</math>
| skewness = <math>\frac{q-p}{\sqrt{npq}}</math>
| skewness = \frac{q-p}{\sqrt{npq}}
| skewness = \frac{q-p}{\sqrt{npq}}
| skewness = frac { q-p }{ sqrt { npq }}
| 偏度 = frac { q-p }{ sqrt { npq }}
| kurtosis = <math>\frac{1-6pq}{npq}</math>
| kurtosis = <math>\frac{1-6pq}{npq}</math>
| kurtosis = \frac{1-6pq}{npq}
| kurtosis = \frac{1-6pq}{npq}
| 峭度 = frac {1-6pq }{ npq }
| 峰度 = frac {1-6pq }{ npq }
| entropy = <math>\frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)</math><br /> in [[Shannon (unit)|shannons]]. For [[nat (unit)|nats]], use the natural log in the log.
| entropy = <math>\frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)</math><br /> in [[Shannon (unit)|shannons]]. For [[nat (unit)|nats]], use the natural log in the log.
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q = 1 − p).
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: success/yes/true/one (with probability p) or failure/no/false/zero (with probability q = 1 − p).
在概率论和统计学中，参数为 n 和 p 的二项分布是 n 个独立实验序列中成功次数的离散概率分布，每个实验询问一个 是-否 问题，每个实验都有自己的布尔值结果: 成功/是/正确/一 (概率为 p)或 失败/否/错误/零 (概率为 q = 1-p)。
A single success/failure experiment is also called a [[Bernoulli trial]] or Bernoulli experiment and a sequence of outcomes is called a [[Bernoulli process]]; for a single trial, i.e., ''n'' = 1, the binomial distribution is a [[Bernoulli distribution]]. The binomial distribution is the basis for the popular [[binomial test]] of [[statistical significance]].
A single success/failure experiment is also called a [[Bernoulli trial]] or Bernoulli experiment and a sequence of outcomes is called a [[Bernoulli process]]; for a single trial, i.e., ''n'' = 1, the binomial distribution is a [[Bernoulli distribution]]. The binomial distribution is the basis for the popular [[binomial test]] of [[statistical significance]].
A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
一个单一的结果为成功/失败的实验也被称为伯努利试验或伯努利实验，一系列伯努利实验结果被称为伯努利过程; 对于一个单一的实验，例如，n = 1，这个二项分布是一个伯努利分布。二项分布是流行统计显著性二项检验的基础。
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.
二项分布经常被用来模拟大小为 n 的样本中的成功数量，这些样本是用 大小为N的种群中的替代物抽取的。如果抽样没有更换，抽样就不是独立的，所以得到的分布是超几何分布，而不是二项分布。然而，对于 N 比 n 大得多的情况，二项分布仍然是一个很好的近似值，并且被广泛使用。
In general, if the random variable X follows the binomial distribution with parameters n ∈ ℕ and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:
In general, if the random variable X follows the binomial distribution with parameters n ∈ ℕ and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n independent Bernoulli trials is given by the probability mass function:
一般来说，如果随机变量 x 服从参数 n ∈ n 和 p ∈[0,1]的二项分布，我们写 x ~ b (n，p)。在 n 个独立的 Bernoulli 试验中获得 k 成功的概率是由美国概率质量函数协会给出的:
一般来说，如果随机变量 x 服从参数 n ∈ N 和 p ∈[0,1]的二项分布，我们写作 x ~ b (n，p)。在 n 个独立的伯努利试验中获得 k 成功的概率是由概率质量函数给出的:
is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows. k successes occur with probability pk and n − k failures occur with probability (1 − p)n − k. However, the k successes can occur anywhere among the n trials, and there are \binom{n}{k} different ways of distributing k successes in a sequence of n trials.
is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows. k successes occur with probability pk and n − k failures occur with probability (1 − p)n − k. However, the k successes can occur anywhere among the n trials, and there are \binom{n}{k} different ways of distributing k successes in a sequence of n trials.
是二项式系数，因而得名。这个公式可以理解为。K次成功发生在概率为 pk 的情况下， n-k 次失败发生在概率为(1-p) n-k 的情况下。然而，k 次成功可以发生在 n 个试验中的任何地方，并且在 n 个试验序列中有不同的 k 次成功的分配方法。
In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as
In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as
在为二项分布概率创建参考表时，通常表中最多填充到 n/2的值。这是因为对于 k > n/2，概率可以通过它的补来计算
and comparing it to 1. There is always an integer M that satisfies
and comparing it to 1. There is always an integer M that satisfies
并与1相比较。总有一个整数 m 满足
并与1相比较。总有一个整数 M 满足
f(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which f is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode.
f(k, n, p) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which f is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable outcome (that is, the most likely, although this can still be unlikely overall) of the Bernoulli trials and is called the mode.
F (k，n，p)对 k < m 是单调递增的，对 k > m 是单调递减的，但(n + 1) p 是整数的情况除外。在这种情况下，有两个值 f 是最大的: (n + 1) p 和(n + 1) p-1。M 是伯努利试验最有可能的结果(也就是说，最有可能的结果，尽管总的来说仍然不太可能) ，被称为模式。
F (k，n，p)对 k < m 是单调递增的，对 k > m 是单调递减的，但(n + 1) p 是整数的情况除外。在这种情况下，有两个值使 f 是最大的: (n + 1) p 和(n + 1) p-1。M 是伯努利试验最有可能的结果(也就是说，最有可能的结果，尽管总的来说仍然存在不发生的情况) ，被称为模。
It can also be represented in terms of the regularized incomplete beta function, as follows:
It can also be represented in terms of the regularized incomplete beta function, as follows:
它也可以用正则化不完全 beta 函数来表示，如下:
Some closed-form bounds for the cumulative distribution function are given below.
Some closed-form bounds for the cumulative distribution function are given below.
下面给出了累积分布函数的一些闭式界。
If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is:
If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is:
如果 x ~ b (n，p) ，即 x 是一个二元分布的随机变量，n 是实验的总数，p 是每个实验产生成功结果的概率，那么 x 的期望值是:
如果 x ~ b (n，p) ，即 x 是一个服从二项分布的随机变量，n 是实验的总数，p 是每个实验产生成功结果的概率，那么 x 的期望值是:
This follows from the linearity of the expected value along with fact that is the sum of identical Bernoulli random variables, each with expected value . In other words, if X_1, \ldots, X_n are identical (and independent) Bernoulli random variables with parameter , then X = X_1 + \cdots + X_n and
This follows from the linearity of the expected value along with fact that is the sum of identical Bernoulli random variables, each with expected value . In other words, if X_1, \ldots, X_n are identical (and independent) Bernoulli random variables with parameter , then X = X_1 + \cdots + X_n and
这是由于两随机变量线性组合的期望值实际上是两个独立的伯努利随机变量的期望的线性组合。换句话说，如果 x1，ldots，xn 是独立的伯努利随机变量，那么 x = x1 + cdots + xn
:<math>\operatorname{E}[X] = \operatorname{E}[X_1 + \cdots + X_n] = \operatorname{E}[X_1] + \cdots + \operatorname{E}[X_n] = p + \cdots + p = np.</math>
:<math>\operatorname{E}[X] = \operatorname{E}[X_1 + \cdots + X_n] = \operatorname{E}[X_1] + \cdots + \operatorname{E}[X_n] = p + \cdots + p = np.</math>
This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.
This similarly follows from the fact that the variance of a sum of independent random variables is the sum of the variances.
这同样也是从这样一个事实出发: 一个独立随机变量之和的方差是方差之和。
这同样也是从这样一个事实出发: 独立随机变量之和的方差是方差之和。
The first 6 central moments are given by
The first 6 central moments are given by
前6个中心矩阵由
:<math>\begin{align}
:<math>\begin{align}
Usually the mode of a binomial B(n, p) distribution is equal to \lfloor (n+1)p\rfloor, where \lfloor\cdot\rfloor is the floor function. However, when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:
Usually the mode of a binomial B(n, p) distribution is equal to \lfloor (n+1)p\rfloor, where \lfloor\cdot\rfloor is the floor function. However, when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:
通常二项式 b (n，p)分布的模式等于 lfloor (n + 1) p 楼层，其中 lfloor cdot 楼层是楼层函数。然而，当(n + 1) p 是整数且 p 既不是0也不是1时，分布有两种模式: (n + 1) p 和(n + 1) p-1。当 p 等于0或1时，模态将相应地为0和 n。这些情况可概述如下:
通常二项式 b (n，p)分布的模式等于 lfloor (n + 1) p 楼层，其中 lfloor cdot 楼层是阶梯函数。然而，当(n + 1) p 是整数且 p 既不是0也不是1时，分布有两种模式: (n + 1) p 和(n + 1) p-1。当 p 等于0或1时，模式将相应地为0和 n。这些情况可概述如下:
==[[用户:南风|南风]]（[[用户讨论:南风|讨论]]）[翻译]存疑“the floor function”
: <math>\text{mode} =
: <math>\text{mode} =
For p=0 only f(0) has a nonzero value with f(0)=1. For p=1 we find f(n)=1 and f(k)=0 for k\neq n. This proves that the mode is 0 for p=0 and n for p=1.
For p=0 only f(0) has a nonzero value with f(0)=1. For p=1 we find f(n)=1 and f(k)=0 for k\neq n. This proves that the mode is 0 for p=0 and n for p=1.
对于 p = 0，只有 f (0)有一个非零值，f (0) = 1。对于 p = 1，我们发现 f (n) = 1和 f (k) = 0对于 k neq n。这证明了 p = 0时模式为0，p = 1时模式为 n。
对于 p = 0，只有 f (0)有一个非零值，f (0) = 1。对于 p = 1，我们发现 f (n) = 1，对于 k neq n， f (k) = 0。这证明了 p = 0时模为0，p = 1时模为 n。
where D(a || p) is the relative entropy between an a-coin and a p-coin (i.e. between the Bernoulli(a) and Bernoulli(p) distribution):
where D(a || p) is the relative entropy between an a-coin and a p-coin (i.e. between the Bernoulli(a) and Bernoulli(p) distribution):
其中 d (a | | p)是一枚硬币和一枚 p 相对熵之间的距离。在伯努利(a)和伯努利(p)分布之间:
其中 d (a | | p)是a-coin，p-coin 相对熵之间的距离。在伯努利(a)和伯努利(p)分布之间:
==~~~[翻译]存疑“a-coin，p-coin”
| pages = 21–25
| pages = 21–25
Asymptotically, this bound is reasonably tight; see
Asymptotically, this bound is reasonably tight; see
从渐近的角度来看，这个约束是相当严格; 参见
}}</ref>
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For p = 1/2 and k ≥ 3n/8 for even n, it is possible to make the denominator constant:
For p = 1/2 and k ≥ 3n/8 for even n, it is possible to make the denominator constant:
对于 p = 1/2和 k ≥3n/8的偶数 n，可以使分母为常数:
对于 p = 1/2且 k ≥3n/8且n为偶数时，可以使分母为常数:
===Tail bounds===
===Tail bounds===
When n is known, the parameter p can be estimated using the proportion of successes: \widehat{p} = \frac{x}{n}. This estimator is found using maximum likelihood estimator and also the method of moments. This estimator is unbiased and uniformly with minimum variance, proven using Lehmann–Scheffé theorem, since it is based on a minimal sufficient and complete statistic (i.e.: x). It is also consistent both in probability and in MSE.
When n is known, the parameter p can be estimated using the proportion of successes: \widehat{p} = \frac{x}{n}. This estimator is found using maximum likelihood estimator and also the method of moments. This estimator is unbiased and uniformly with minimum variance, proven using Lehmann–Scheffé theorem, since it is based on a minimal sufficient and complete statistic (i.e.: x). It is also consistent both in probability and in MSE.
当 n 已知时，参数 p 可以使用成功的比例来估计: widehat { p } = frac { x }{ n }。利用极大似然估计和矩方法求出了该估计量。利用 Lehmann-scheffé 定理证明了该估计量的无偏一致最小方差，因为该估计量是基于一个极小充分完全统计量(即:。: x).它在概率和均方误差方面也是一致的。
当 n 已知时，参数 p 可以使用成功的比例来估计: widehat { p } = frac { x }{ n }。利用'''<font color="#ff8000">极大似然估计</font>'''和'''<font color="#ff8000">矩方法</font>'''求出了该估计量。利用 '''<font color="#ff8000">Lehmann-scheffé 定理</font>'''证明了该估计量的'''<font color="#ff8000">无偏一致最小方差</font>'''，因为该估计量是基于一个极小充分完全统计量(即:。: x).它在概率和均方误差方面也是一致的。
A closed form Bayes estimator for p also exists when using the Beta distribution as a conjugate prior distribution. When using a general \operatorname{Beta}(\alpha, \beta) as a prior, the posterior mean estimator is: \widehat{p_b} = \frac{x+\alpha}{n+\alpha+\beta}. The Bayes estimator is asymptotically efficient and as the sample size approaches infinity (n → ∞), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability.
A closed form Bayes estimator for p also exists when using the Beta distribution as a conjugate prior distribution. When using a general \operatorname{Beta}(\alpha, \beta) as a prior, the posterior mean estimator is: \widehat{p_b} = \frac{x+\alpha}{n+\alpha+\beta}. The Bayes estimator is asymptotically efficient and as the sample size approaches infinity (n → ∞), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability.
利用 Β分布作为共轭先验分布时，p 的 Bayes 估计也是一个封闭形式。当使用一个通用算子名{ Beta }(alpha，Beta)作为先验时，后验平均估计量为: widehat { p _ b } = frac { x + alpha }{ n + alpha + Beta }。贝叶斯估计是渐近有效的，当样本容量接近无穷大(n →∞)时，它逼近最大似然估计解。贝叶斯估计是有偏的(多少取决于先验) ，可容许和一致的概率。
利用 beta分布作为'''<font color="#ff8000">共轭先验分布</font>'''时，也存在p的封闭形式 Bayes 估计。当使用一个通用算子名{ Beta }(alpha，Beta)作为先验时，后验平均估计量为: widehat { p _ b } = frac { x + alpha }{ n + alpha + Beta }。贝叶斯估计是渐近有效的，当样本容量接近无穷大(n →∞)时，它逼近最大似然估计解。贝叶斯估计是有偏的(多少取决于先验) ，可接受且一致的概率。
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed.
即使对于非常大的 n 值，均值的实际分布也是非常非正态的。针对这一问题，提出了几种估计置信区间的方法。
即使对于非常大的 n 值，均值的实际分布也是非正态的。针对这一问题，提出了几种估计置信区间的方法。
One can also obtain ''lower'' bounds on the tail <math>F(k;n,p) </math>, known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that<ref>{{cite book |author1=Robert B. Ash |title=Information Theory |url=https://archive.org/details/informationtheor00ashr |url-access=limited |date=1990 |publisher=Dover Publications |page=[https://archive.org/details/informationtheor00ashr/page/n81 115]}}</ref>
One can also obtain ''lower'' bounds on the tail <math>F(k;n,p) </math>, known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that<ref>{{cite book |author1=Robert B. Ash |title=Information Theory |url=https://archive.org/details/informationtheor00ashr |url-access=limited |date=1990 |publisher=Dover Publications |page=[https://archive.org/details/informationtheor00ashr/page/n81 115]}}</ref>
A continuity correction of 0.5/n may be added.
A continuity correction of 0.5/n may be added.
可以加上0.5/n 的连续性修正。
可以加上0.5/n 的连续校正。
=== Estimation of parameters ===
=== Estimation of parameters ===
For example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).
For example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).
例如，想象一下把 n 个球扔到一个篮子里，然后把击中的球扔到另一个篮子里。如果 p 是命中 UX 的概率，那么 x ~ b (n，p)是命中 UX 的球数。如果 q 是击中 yy 的概率，那么击中 yy 的球数是 y ~ b (x，q) ，因此 y ~ b (n，pq)。
例如，想象一下把 n 个球扔到一个篮子UX里，然后把击中的球扔到另一个篮子UY里。如果 p 是命中 UX 的概率，那么 x ~ b (n，p)是命中 UX 的球数。如果 q 是击中 UY 的概率，那么击中 UY1 的球数是 y ~ b (x，q) ，因此 y ~ b (n，pq)。
The notation in the formula below differs from the previous formulas in two respects:<ref name="Wilson1927">{{Citation |last = Wilson |first=Edwin B. |date = June 1927 |title = Probable inference, the law of succession, and statistical inference |url = http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |journal = Journal of the American Statistical Association |volume=22 |issue=158 |pages=209–212 |access-date= 2015-01-05 |doi = 10.2307/2276774 |url-status=dead |archive-url = https://web.archive.org/web/20150113082307/http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |archive-date = 2015-01-13 |jstor = 2276774 }}</ref>
The notation in the formula below differs from the previous formulas in two respects:<ref name="Wilson1927">{{Citation |last = Wilson |first=Edwin B. |date = June 1927 |title = Probable inference, the law of succession, and statistical inference |url = http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |journal = Journal of the American Statistical Association |volume=22 |issue=158 |pages=209–212 |access-date= 2015-01-05 |doi = 10.2307/2276774 |url-status=dead |archive-url = https://web.archive.org/web/20150113082307/http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf |archive-date = 2015-01-13 |jstor = 2276774 }}</ref>
Notice that the sum (in the parentheses) above equals (p - pq + 1 - p)^{n-m} by the binomial theorem. Substituting this in finally yields
Notice that the sum (in the parentheses) above equals (p - pq + 1 - p)^{n-m} by the binomial theorem. Substituting this in finally yields
注意，上面的和(在括号中)等于(p-pq + 1-p) ^ { n-m }二项式定理。用最终产量来代替这一点
注意，上面的和(在括号中)等于(p-pq + 1-p) ^ { n-m }二项式定理。最终将其替换为
| year = 2012
| year = 2012
and thus Y \sim B(n, pq) as desired.
and thus Y \sim B(n, pq) as desired.
因此 y sim b (n，pq)为所需。
因此 y sim b (n，pq)为所需值。
The exact ([[Binomial proportion confidence interval#Clopper–Pearson interval|Clopper–Pearson]]) method is the most conservative.<ref name="Brown2001" />
The exact ([[Binomial proportion confidence interval#Clopper–Pearson interval|Clopper–Pearson]]) method is the most conservative.<ref name="Brown2001" />
If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(n, p) is given by the normal distribution
If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(n, p) is given by the normal distribution
如果 n 足够大，那么分布的倾斜就不会太大。在这种情况下，通过正态分布给出 b (n，p)的合理近似
如果 n 足够大，那么分布的偏斜就不会太大。在这种情况下，通过正态分布给出 b (n，p)的合理近似
The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed or at least p tends to zero. Therefore, the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10.
The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed or at least p tends to zero. Therefore, the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10.
当试验数量趋于无穷大，而产品 np 保持不变或者至少 p 趋于零时，二项分布收敛到泊松分佈。因此，参数 λ = np 的泊松分佈可以作为二项分布 b (n，p)的近似，如果 n 是足够大，p 足够小的话。根据两个经验法则，如果 n ≥20和 p ≤0.05，或者如果 n ≥100和 np ≤10，这个近似是好的。
当试验数量趋于无穷大，而产品 np 保持不变或者至少 p 趋于零时，二项分布收敛到泊松分佈。因此，参数 λ = np 的泊松分布可以作为二项分布 b (n，p)的近似，如果 n 是足够大，p 足够小的话。根据两个经验法则，如果 n ≥20和 p ≤0.05，或者如果 n ≥100和 np ≤10，这个近似是好的。
Given a uniform prior, the posterior distribution for the probability of success given independent events with observed successes is a beta distribution.
Given a uniform prior, the posterior distribution for the probability of success given independent events with observed successes is a beta distribution.
给定一个统一的先验，给定观察到成功结果的独立事件成功概率的后验分布是一个 beta分布。
\end{align}</math>
\end{align}</math>
Methods for random number generation where the marginal distribution is a binomial distribution are well-established.
Methods for random number generation where the marginal distribution is a binomial distribution are well-established.
'''<font color="#ff8000">边缘分布</font>'''是二项分布的随机数产生方法已经很完善了。
:<math>\begin{align}
:<math>\begin{align}
One way to generate random samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that for all values from through . (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a pseudorandom number generator to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.
One way to generate random samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that for all values from through . (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a pseudorandom number generator to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.
一种从二项分布中产生随机样本的方法是使用'''<font color="#ff8000">反演算法</font>'''。要做到这一点，我们必须计算从到的所有值的概率。(为了包含整个样本空间，这些概率之和应该接近于一。)然后，通过使用伪随机数生成器来生成数值为0或1的样本，人们可以使用在第一步计算出的概率将计算出的样本转换成离散数。
&= \binom{n}{m} (pq)^m \left( \sum_{k=m}^n \binom{n-m}{k-m} \left(p(1-q)\right)^{k-m} (1-p)^{n-k} \right)
&= \binom{n}{m} (pq)^m \left( \sum_{k=m}^n \binom{n-m}{k-m} \left(p(1-q)\right)^{k-m} (1-p)^{n-k} \right)
===Bernoulli distribution===
===Bernoulli distribution===
伯努利分布
The [[Bernoulli distribution]] is a special case of the binomial distribution, where ''n'' = 1. Symbolically, ''X'' ~ B(1, ''p'') has the same meaning as ''X'' ~ Bernoulli(''p''). Conversely, any binomial distribution, B(''n'', ''p''), is the distribution of the sum of ''n'' [[Bernoulli trials]], Bernoulli(''p''), each with the same probability ''p''.<ref>{{cite web|last1=Taboga|first1=Marco|title=Lectures on Probability Theory and Mathematical Statistics|url=https://www.statlect.com/probability-distributions/binomial-distribution#hid3|website=statlect.com|accessdate=18 December 2017}}</ref>
The [[Bernoulli distribution]] is a special case of the binomial distribution, where ''n'' = 1. Symbolically, ''X'' ~ B(1, ''p'') has the same meaning as ''X'' ~ Bernoulli(''p''). Conversely, any binomial distribution, B(''n'', ''p''), is the distribution of the sum of ''n'' [[Bernoulli trials]], Bernoulli(''p''), each with the same probability ''p''.<ref>{{cite web|last1=Taboga|first1=Marco|title=Lectures on Probability Theory and Mathematical Statistics|url=https://www.statlect.com/probability-distributions/binomial-distribution#hid3|website=statlect.com|accessdate=18 December 2017}}</ref>
===Poisson binomial distribution===
===Poisson binomial distribution===
泊松二项分布
The binomial distribution is a special case of the [[Poisson binomial distribution]], or [[general binomial distribution]], which is the distribution of a sum of ''n'' independent non-identical [[Bernoulli trials]] B(''p<sub>i</sub>'').<ref>
The binomial distribution is a special case of the [[Poisson binomial distribution]], or [[general binomial distribution]], which is the distribution of a sum of ''n'' independent non-identical [[Bernoulli trials]] B(''p<sub>i</sub>'').<ref>
===Normal approximation===
===Normal approximation===
正态逼近
Category:Factorial and binomial topics
Category:Factorial and binomial topics