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。
  • If, in a particular situation, a random event has only two possible results, its probability distribution is a binomial distribution, expressed as .

• 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（ 且 ），二项分布可以近似为正态分布。
  • If n is large enough ( and ), the binomial distribution can be approximately regarded as a normal distribution.

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）：

B 的概率（Probability of B）：

A 与 B 的概率满足 。

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

二项分布的均值计算公式为：
The mean of the binomial distribution is calculated as .

二项分布的标准差计算公式为：
The standard deviation of the binomial distribution is calculated as .

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 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 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.