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| pdf_image = Probability mass function for the binomial distribution
| pdf_image = Probability mass function for the binomial distribution
2012年3月24日 | pdf 图片 = 概率质量函数二项分布
2012年3月24日 | pdf 图片 = '''<font color="#ff8000">概率质量函数二项分布</font>'''
| cdf_image = [[File:Binomial distribution cdf.svg|300px|Cumulative distribution function for the binomial distribution]]
| cdf_image = [[File:Binomial distribution cdf.svg|300px|Cumulative distribution function for the binomial distribution]]
| cdf_image = Cumulative distribution function for the binomial distribution
| cdf_image = Cumulative distribution function for the binomial distribution
2012年3月24日 | cdf 图像 = 累积分布函数二项分布
2012年3月24日 | cdf 图像 = '''<font color="#ff8000">累积分布函数二项分布</font>'''
| notation = <math>B(n,p)</math>
| notation = <math>B(n,p)</math>
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)。
在概率论和统计学中，参数为 n 和 p 的'''<font color="#ff8000">二项分布</font>'''是 n 个独立实验序列中成功次数的'''<font color="#ff8000">离散概率分布</font>'''，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.
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。
一个单一的结果为成功/失败的实验也被称为'''<font color="#ff8000">伯努利试验</font>'''或伯努利实验，一系列伯努利实验结果被称为'''<font color="#ff8000">伯努利过程</font>'''; 对于一个单一的实验，例如，n = 1，这个二项分布是一个'''<font color="#ff8000">伯努利分布</font>'''。二项分布是流行'''<font color="#ff8000">统计显著性</font>''''''<font color="#ff8000">二项检验</font>'''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。
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'''<font color="#ff8000">超几何分布</font>'''，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'''<font color="#ff8000">随机变量</font>''' 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'''<font color="#ff8000">概率质量函数</font>'''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'''<font color="#ff8000">二项式系数</font>'''，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。
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。
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'''<font color="#ff8000">模</font>'''。
Suppose a biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is
Suppose a biased coin comes up heads with probability 0.3 when tossed. The probability of seeing exactly 4 heads in 6 tosses is
假设抛出一枚'''<font color="#ff8000">有偏硬币</font>'''时，正面朝上的概率为0.3。在6次抛掷中正好看到4个正面的概率是
The cumulative distribution function can be expressed as:
The cumulative distribution function can be expressed as:
'''<font color="#ff8000">累积分布函数</font>'''可以表达为:
where \lfloor k\rfloor is the "floor" under k, i.e. the greatest integer less than or equal to k.
where \lfloor k\rfloor is the "floor" under k, i.e. the greatest integer less than or equal to k.
是 k 下面的”楼层” ，也就是。小于或等于 k 的最大整数。
是 k 下面的”楼层” ，也就是。小于或等于 k 的'''<font color="#ff8000">最大整数</font>'''。
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'''<font color="#ff8000">正则化不完全 beta 函数</font>'''来表示，如下:
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:
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'''<font color="#ff8000">期望值</font>'''是:
The variance is:
The variance is:
'''<font color="#ff8000">方差</font>'''是:
:<math> \operatorname{Var}(X) = np(1 - p).</math>
:<math> \operatorname{Var}(X) = np(1 - p).</math>
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)分布的模等于 lfloor (n + 1) p 楼层，其中 lfloor cdot 楼层是下限函数。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: (n + 1) p 和(n + 1) p-1。当 p 等于0或1时，模将相应地为0和 n。这些情况可概述如下:
通常二项式 b (n，p)分布的'''<font color="#ff8000">模</font>'''等于 lfloor (n + 1) p 楼层，其中 lfloor cdot 楼层是'''<font color="#ff8000">下限函数</font>'''。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: (n + 1) p 和(n + 1) p-1。当 p 等于0或1时，模将相应地为0和 n。这些情况可概述如下:
In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:
In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:
一般来说，没有单一的公式可以找到一个二项分布的'''<font color="#ff8000">中位数</font>'''，甚至可能是非唯一的。然而，已经确立了若干特别成果:
* If ''np'' is an integer, then the mean, median, and mode coincide and equal ''np''.<ref>{{cite journal|last=Neumann|first=P.|year=1966|title=Über den Median der Binomial- and Poissonverteilung|journal=Wissenschaftliche Zeitschrift der Technischen Universität Dresden|volume=19|pages=29–33|language=German}}</ref><ref>Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", [[The Mathematical Gazette]] 94, 331-332.</ref>
* If ''np'' is an integer, then the mean, median, and mode coincide and equal ''np''.<ref>{{cite journal|last=Neumann|first=P.|year=1966|title=Über den Median der Binomial- and Poissonverteilung|journal=Wissenschaftliche Zeitschrift der Technischen Universität Dresden|volume=19|pages=29–33|language=German}}</ref><ref>Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", [[The Mathematical Gazette]] 94, 331-332.</ref>