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第1行： − + − + − |1=zh-hant:參數;zh-cn:参数;zh-tw:母數+ − |2= zh-cn:泊松; zh-tw:卜瓦松; zh-hk:泊松;+ − |3=zh-hant:二項分布;zh-tw:二項式分布;zh-cn:二项分布+ − |4= zh-hans:矩; zh-tw:動差;zh-hant:矩+ − }}+ − + − + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + − + − + − + − + − + − + − + − + − + − + − + − + − + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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{{NoteTA|G1=Math
{{short description|Probability distribution}}
|T=zh-tw:二項式分布;zh-cn:二项分布
{{Redirect|Binomial model|the binomial model in options pricing|Binomial options pricing model}}
{{see also|Negative binomial distribution}}
<!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one.
{{Infobox 機率分佈
|name =二項分布
<!-- EDITORS! Please see Wikipedia:WikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one.
|type =質量
< ! – 本文编辑，参见讨论概率分布使用标准的文章[[Wikipedia: WikiProject Probability # standards]]。
-->
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{{Probability distribution
{{Probability distribution
<font color="#ff8000">概率分布 Probability distribution </font>
| name = Binomial distribution
| name = Binomial distribution
名称 = <font color="#ff8000">二项分布 Binomial distribution </font>
| type = mass
| type = mass
类型 = 质量，这里指<font color="#ff8000">离散型 discrete</font>
| pdf_image = [[File:Binomial distribution pmf.svg|300px|Probability mass function for the binomial distribution]]
| pdf_image = [[File:Binomial distribution pmf.svg|300px|Probability mass function for the binomial distribution]]
| pdf_image = Probability mass function for the binomial distribution
| 概率质量函数图像 = '''<font color="#ff8000">二项分布的概率质量函数 Probability mass function for the binomial distribution </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]]
| notation = ''B''(''n'', ''p'')
|parameters =<math>n \geq 0</math> 试验次数 ([[整数]])<br /><math>0\leq p \leq 1</math> 成功概率 ([[实数]])
| cdf_image = Cumulative distribution function for the binomial distribution
|support =<math>k \in \{0,\dots,n\}\!</math>
|pdf =<math>{n\choose k} p^k (1-p)^{n-k} \!</math>
| 累积分布函数图像 = '''<font color="#ff8000">二项分布的累积分布函数 Cumulative distribution function for the binomial distribution </font>'''
|cdf =<math>I_{1-p}(n-\lfloor k\rfloor, 1+\lfloor k\rfloor) \!</math>
| notation = <math>B(n,p)</math>
|mean =<math>n\,p\!</math>
|median =<math>\{\lfloor np\rfloor, \lceil (n+1)p \rceil\}</math>之一
| notation = B(n,p)
|mode =<math>\lfloor (n+1)\,p\rfloor\!</math>或<math>\lfloor (n+1)\,p\rfloor\!-1</math>
|variance =<math>n\,p\,(1-p)\!</math>
| 符号 = <math>B(n,p)</math>
|skewness =<math>\frac{1-2\,p}{\sqrt{n\,p\,(1-p)}}\!</math>
|kurtosis =<math>\frac{1-6\,p\,(1-p)}{n\,p\,(1-p)}\!</math>
| parameters = <math>n \in \{0, 1, 2, \ldots\}</math> – number of trials<br /><math>p \in [0,1]</math> – success probability for each trial<br /><math>q = 1 - p</math>
|entropy =<math>\frac{1}{2} \ln \left( 2 \pi n e p (1-p) \right) + O \left( \frac{1}{n} \right)\!</math>
|mgf =<math>(1-p + p\,e^t)^n \!</math>
| parameters = n \in \{0, 1, 2, \ldots\} – number of trials<br />p \in [0,1] – success probability for each trial<br />q = 1 - p
|char =<math>(1-p + p\,e^{i\,t})^n \!</math>
| 参数 = <br /><math>n \in \{0, 1, 2, \ldots\}</math> – --- 试验次数； <br /><math>p \in [0,1]</math> – -- 每个试验的成功概率； <br /><math>q = 1 - p</math>
| support = <math>k \in \{0, 1, \ldots, n\}</math> – number of successes
| support = k \in \{0, 1, \ldots, n\} – number of successes
| 支持 = <br /><math>k \in \{0, 1, \ldots, n\}</math> – --- 成功的数量
| pdf = <math>\binom{n}{k} p^k q^{n-k}</math>
| pdf = \binom{n}{k} p^k q^{n-k}
|<font color="#ff8000">概率质量函数 Probability mass function </font> = <math>\binom{n}{k} p^k q^{n-k}</math>
| cdf = <math>I_{q}(n - k, 1 + k)</math>
| cdf = I_{q}(n - k, 1 + k)
| <font color="#ff8000">累积分布函数 Cumulative distribution function </font> = <math>I_{q}(n - k, 1 + k)</math>
| mean = <math>np</math>
| mean = np
<font color="#ff8000">平均值 mean</font> = <math>np</math>
| median = <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math>
| median = \lfloor np \rfloor or \lceil np \rceil
<font color="#ff8000">中位数 median</font> = <math>\lfloor np \rfloor</math> 或 <math>\lceil np \rceil</math>
| mode = <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math>
| mode = \lfloor (n + 1)p \rfloor or \lceil (n + 1)p \rceil - 1
| <font color="#ff8000">模 mode</font> = <math>\lfloor (n + 1)p \rfloor</math> 或 <math>\lceil (n + 1)p \rceil - 1</math>
| variance = <math>npq</math>
| variance = npq
| <font color="#ff8000">方差 variance</font> = <math>npq</math>
| skewness = <math>\frac{q-p}{\sqrt{npq}}</math>
| skewness = \frac{q-p}{\sqrt{npq}}
| <font color="#ff8000">偏度 skewness</font> = <math>\frac{q-p}{\sqrt{npq}}</math>
| kurtosis = <math>\frac{1-6pq}{npq}</math>
| kurtosis = \frac{1-6pq}{npq}
| <font color="#ff8000">峰度 kurtosis</font> = <math>\frac{1-6pq}{npq}</math>
| 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 = \frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)<br /> in shannons. For nats, use the natural log in the log.
| <font color="#ff8000">熵 entropy</font> = <math>\frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)</math>用<font color="#ff8000">香农熵 Shannon entropy</font>测量。对于<font color="#ff8000">分布式消息队列系统 NATS </font>，使用日志中的自然日志。
| mgf = <math>(q + pe^t)^n</math>
| mgf = (q + pe^t)^n
| <font color="#ff8000">矩量母函数 Moment Generating Function</font> = <math>(q + pe^t)^n</math>
| char = <math>(q + pe^{it})^n</math>
| char = (q + pe^{it})^n
| <font color="#ff8000">特征函数 characteristic function</font> = <math>(q + pe^{it})^n</math>
| pgf = <math>G(z) = [q + pz]^n</math>
| pgf = G(z) = [q + pz]^n
| <font color="#ff8000">概率母函数 probability generating function</font> = <math>G(z) = [q + pz]^n</math>
| fisher = <math> g_n(p) = \frac{n}{pq} </math><br />(for fixed <math>n</math>)
| fisher = g_n(p) = \frac{n}{pq} <br />(for fixed n)
| <font color="#ff8000">费雪信息量 fisher information</font> = <math> g_n(p) = \frac{n}{pq} </math><br />(对于固定的 <math>n</math>)
}}
}}
}}