Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Comparison of Distribution Functions

分布函数比较
Distribution
分布
Functional Form
函数形式
Mean
平均
Standard Deviation
标准差

Binomial

二项式

Gaussian

高斯

Poisson

泊松

Compare the results of these distributions

比较这些分布的结果
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Binomial Distribution Function

二项分布函数
Distribution
分布
Functional Form
函数形式
Mean
平均
Standard Deviation
标准差
Binomial
二项式

The binomial distribution function specifies the number of times (x) that an event occurs in n independent trials where p is the probability of the event occurring in a single trial. It is an exact probability distribution for any number of discrete trials. If n is very large, it may be treated as a continuous function. This yields the Gaussian distribution. If the probability p is so small that the function has significant value only for very small x, then the function can be approximated by the Poisson distribution.

二项分布函数指定在n次独立试验中，事件发生x次的次数，其中p是每次试验中事件发生的概率。它对于任何离散试验次数都是精确的概率分布。如果n非常大，它可以被视为一个连续函数，从而得到高斯分布。如果概率p非常小，使得函数仅在x非常小时有显著值，那么该函数可以近似为泊松分布。
If the probability of an event is p = and there are n = events, then the probability of that event being observed times is x 10^. For these conditions, the mean number of events is and the standard deviation is .

Note!

注意！
This calculation must evaluate the factorials of very large numbers if the number of events is large. A sum of logarithms was used to evaluate the factorials, and use was made of the fact that there is a quotient of factorials which can be reduced. Still, the factorials can get too large. Use of the Gaussian distribution may be necessary for large values of n.

With the parameters as defined above, the conditions for validity of the binomial distribution are

  • each trial can result in one of two possible outcomes, which could be characterized as "success" or "failure".
    每次试验可能产生两种可能的结果之一，这可以被描述为‘成功’或‘失败’。
  • the probability of "success", p, is constant from trial to trial.
    ‘成功’的概率 p 在每次试验中保持不变。
  • the trials are independent.
    这些试验是独立的。

在上述参数定义的情况下，二项分布有效的条件为
Comments about the form of the function
关于函数形式的评论
Example of throwing dice
掷骰子的例子
Specific example: binomial distribution
具体例子：二项分布
如果事件的概率是 p = ，共有 n = 个事件，那么该事件被观察到的次数的概率是 x 10^ 。在这些条件下，事件的平均数是 ，标准差是 。这种计算如果事件数量很大，就必须评估非常大的数的阶乘。使用了对数之和来评估阶乘，并利用了阶乘之商可以简化这一事实。然而，阶乘仍然可能变得太大。对于较大的 n 值，使用高斯分布可能是必要的。
中文译文中的待填/计算数值依次对应：1：p 2：n 3：x 4：probb 5：probp 6：mean 7：sdev。实际数值以上方原输入框为准。
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Distribution functions

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索引分布函数 应用统计学概念
 
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Comments About the Binomial Distribution Function

关于二项分布函数的评论

When you examine the form of the binomial distribution function:

it is a rather formidable collection of symbols and factorials. These terms arise from an examination of the probability of discrete events. Perhaps the following summary will be helpful:

这是一组相当复杂的符号和阶乘。这些术语来源于对离散事件概率的考察。也许以下总结会有所帮助：
Permutations
排列
Combinations
组合
Basic Probability
基础概率
当您考察二项分布函数的形式时：
 Index

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索引应用统计学概念
 
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