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|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
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{{Probability distribution
| name = Binomial distribution
| type = mass
| pdf_image = [[File:Binomial distribution pmf.svg|300px|Probability mass function for the binomial distribution]]
| cdf_image = [[File:Binomial distribution cdf.svg|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}}
{{short description|Probability distribution}}