====== 二项分布 (Binomial Distribution) ======

  * **定义**：如果在某种特定的情境下，一个随机事件只有两种可能的结果，其概率分布就是一个二项分布，表示为{{:第四章:4.4:bnp.svg}}。
    * If, in a particular situation, a random event has only two possible results, its probability distribution is a binomial distribution, expressed as {{:第四章:4.4:bnp.svg}}.

  * **例子**：投掷硬币得到正面或反面，人的生或死，六面骰子的点数为奇数或偶数，某天下雨还是不下雨。
    * Toss a coin to get heads or tails, a person's life or death, six-sided dice points are odd or even, one day it rains or not.

  * **近似**：如果n足够大（{{:第四章:4.4:pn10.svg}} 且 {{:第四章:4.4:qn10.svg}}），二项分布可以近似为正态分布。
    * If n is large enough ({{:第四章:4.4:pn10.svg}} and {{:第四章:4.4:qn10.svg}}), the binomial distribution can be approximately regarded as a normal distribution.


===== 二项分布的概率 =====
二项分布中总是由两个对立的类目构成：A和B
  *The binomial distribution always consists of two opposing classes, denoted A and B.
  
A 的概率（Probability of A）：{{:第四章:4.4:ppa.svg}}

B 的概率（Probability of B）：{{:第四章:4.4:qpb.svg}}

A 与 B 的概率满足 {{:第四章:4.4:pq1.svg}}。

样本中所包含个体的数目（number of sample）：n

样本中事件类目A发生的数目（number of event）：X

二项分布表达了与从X=0到X=n的每一个X值有关的概率
  *The binomial distribution expresses the probability associated with each value of X from X=0 to X=n
    
    
===== 二项分布的均值和标准差 (Mean and standard deviation of the binomial distribution) =====

二项分布的均值计算公式为：{{:第四章:4.4:mupn.svg}}\\
The mean of the binomial distribution is calculated as {{:第四章:4.4:mupn.svg}}.

二项分布的标准差计算公式为：{{:第四章:4.4:sigmasqrtnpq.svg}}\\
The standard deviation of the binomial distribution is calculated as {{:第四章:4.4:sigmasqrtnpq.svg}}.
    
===== 利用正态分布表求二项分布的概率 (Finding the probability of a binomial distribution using a normal distribution table) =====

此时使用的是连续型分布来估计离散型分布的值，正态分布中的 {{:第四章:4.4:x.svg}} 值是一段，而非一点，当二项分布近似为正态分布时，需要考虑精确上下限。\\
At this point a continuous type distribution is used to estimate the value of a discrete distribution; the value of {{:第四章:4.4:x.svg}} in a normal distribution is a segment, not a point, and exact upper and lower bounds need to be considered when the binomial distribution is approximated as a normal distribution.