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第3行： − {{Probability distribution − | name = Binomial distribution − | type = mass − | pdf_image = [[File:Binomial distribution pmf.svg.png|300px|Probability mass function for the binomial distribution]] − | cdf_image = [[File:Binomial distribution cdf.svg.png|300px|Cumulative distribution function for the binomial distribution]] − | notation = <math>B(n,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> − | support = <math>k \in \{0, 1, \ldots, n\}</math> – number of successes − | pdf = <math>\binom{n}{k} p^k q^{n-k}</math> − | cdf = <math>I_{q}(n - k, 1 + k)</math> − | mean = <math>np</math> − | median = <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math> − | mode = <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math> − | variance = <math>npq</math> − | skewness = <math>\frac{q-p}{\sqrt{npq}}</math> − | kurtosis = <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. − | mgf = <math>(q + pe^t)^n</math> − | char = <math>(q + pe^{it})^n</math> − | pgf = <math>G(z) = [q + pz]^n</math> − | fisher = <math> g_n(p) = \frac{n}{pq} </math><br />(for fixed <math>n</math>) − }} − {{Probability fundamentals}} − {{short description|Probability distribution}} − + − + − {{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.+ − + − <!-- EDITORS! Please see Wikipedia:WikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one.+ − + − < ! – 本文编辑，参见讨论概率分布使用标准的文章[[Wikipedia: WikiProject Probability # standards]]。+ − + − -->+ − + − -->+ − + − -->+ − + − {{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.png|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.png|300px|Cumulative distribution function for the binomial distribution]]+ − + − | cdf_image = Cumulative distribution function for the binomial distribution+ − + − | 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>'''+ − | notation = <math>B(n,p)</math>+ − + − | notation = B(n,p)+ − − | 符号 = <math>B(n,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> − − | parameters = n \in \{0, 1, 2, \ldots\} – number of trials<br />p \in [0,1] – success probability for each trial<br />q = 1 - p − − − − | support = <math>k \in \{0, 1, \ldots, n\}</math> – number of successes − − | support = k \in \{0, 1, \ldots, n\} – number of successes − − − − | pdf = <math>\binom{n}{k} p^k q^{n-k}</math> − − | pdf = \binom{n}{k} p^k q^{n-k} − − − − | cdf = <math>I_{q}(n - k, 1 + k)</math> − − | cdf = I_{q}(n - k, 1 + k) − − − − | mean = <math>np</math> − − | mean = np − − − − | median = <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math> − − | median = \lfloor np \rfloor or \lceil np \rceil − − − − | 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 − − − − | variance = <math>npq</math> − − | variance = npq − − − − | skewness = <math>\frac{q-p}{\sqrt{npq}}</math> − − | skewness = \frac{q-p}{\sqrt{npq}} − − − − | kurtosis = <math>\frac{1-6pq}{npq}</math> − − | kurtosis = \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 = \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. − − − − | mgf = <math>(q + pe^t)^n</math> − − | mgf = (q + pe^t)^n − − − − | char = <math>(q + pe^{it})^n</math> − − | char = (q + pe^{it})^n − − − − | pgf = <math>G(z) = [q + pz]^n</math> − − | pgf = G(z) = [q + pz]^n − − − − | 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) − − − − }} − − }}
无编辑摘要
|description=离散分布,阶乘和二项式主题,共轭先验分布,正态逼近,指数族分布
|description=离散分布,阶乘和二项式主题,共轭先验分布,正态逼近,指数族分布
}}
}}
{{Redirect|Binomial model|the binomial model in options pricing|Binomial options pricing model}}
{| class="wikitable" style="float:right; margin-left: 10px;"
|-
| 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>
|-
| 支持
| <br /><math>k \in \{0, 1, \ldots, n\}</math> – --- 成功的数量
|-
|<font color="#ff8000">概率质量函数 </font>
|<math>\binom{n}{k} p^k q^{n-k}</math>
|-
| <font color="#ff8000">累积分布函数 </font>
| <math>I_{q}(n - k, 1 + k)</math>
|-
| <font color="#ff8000">平均值</font> =
| <math>np</math>
|-
| <font color="#ff8000">中位数</font>
| <math>\lfloor np \rfloor</math> 或 <math>\lceil np \rceil</math>
|-
| <font color="#ff8000">模</font>
| <math>\lfloor (n + 1)p \rfloor</math> 或 <math>\lceil (n + 1)p \rceil - 1</math>
|-
| <font color="#ff8000">方差</font>
| <math>npq</math>
|-
| <font color="#ff8000">偏度</font>
| <math>\frac{q-p}{\sqrt{npq}}</math>
|-
| <font color="#ff8000">峰度</font>
| <math>\frac{1-6pq}{npq}</math>
|-
| <font color="#ff8000">熵</font>
| <math>\frac{1}{2} \log_2 (2\pi enpq) + O \left( \frac{1}{n} \right)</math>
|-
| <font color="#ff8000">矩量母函数</font>
| <math>(q + pe^t)^n</math>
|-
| <font color="#ff8000">特征函数</font> =
| <math>(q + pe^{it})^n</math>
|-
| <font color="#ff8000">概率母函数</font>
| <math>G(z) = [q + pz]^n</math>
|-
| <font color="#ff8000">费雪信息量</font>
| <math> g_n(p) = \frac{n}{pq} </math><br />(对于固定的 <math>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>
| 支持 = <br /><math>k \in \{0, 1, \ldots, n\}</math> – --- 成功的数量
|<font color="#ff8000">概率质量函数 Probability mass function </font> = <math>\binom{n}{k} p^k q^{n-k}</math>
| <font color="#ff8000">累积分布函数 Cumulative distribution function </font> = <math>I_{q}(n - k, 1 + k)</math>
<font color="#ff8000">平均值 mean</font> = <math>np</math>
<font color="#ff8000">中位数 median</font> = <math>\lfloor np \rfloor</math> 或 <math>\lceil np \rceil</math>
| <font color="#ff8000">模 mode</font> = <math>\lfloor (n + 1)p \rfloor</math> 或 <math>\lceil (n + 1)p \rceil - 1</math>
| <font color="#ff8000">方差 variance</font> = <math>npq</math>
| <font color="#ff8000">偏度 skewness</font> = <math>\frac{q-p}{\sqrt{npq}}</math>
| <font color="#ff8000">峰度 kurtosis</font> = <math>\frac{1-6pq}{npq}</math>
| <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。
| <font color="#ff8000">矩量母函数 Moment Generating Function</font> = <math>(q + pe^t)^n</math>
| <font color="#ff8000">特征函数 characteristic function</font> = <math>(q + pe^{it})^n</math>
| <font color="#ff8000">概率母函数 probability generating function</font> = <math>G(z) = [q + pz]^n</math>
| <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>)
[[File:Pascal's triangle; binomial distribution.svg.png|thumb|280px|Binomial distribution for <math>p=0.5</math><br />with ''n'' and ''k'' as in [[Pascal's triangle]]<br /><br />The probability that a ball in a [[Bean machine|Galton box]] with 8 layers (''n'' = 8) ends up in the central bin (''k'' = 4) is <math>70/256</math>.|链接=Special:FilePath/Pascal's_triangle;_binomial_distribution.svg.png]]
[[File:Pascal's triangle; binomial distribution.svg.png|thumb|280px|Binomial distribution for <math>p=0.5</math><br />with ''n'' and ''k'' as in [[Pascal's triangle]]<br /><br />The probability that a ball in a [[Bean machine|Galton box]] with 8 layers (''n'' = 8) ends up in the central bin (''k'' = 4) is <math>70/256</math>.|链接=Special:FilePath/Pascal's_triangle;_binomial_distribution.svg.png]]