二项分布的正态近似若干条件

Conditions for Normal Approximation of Binomial Distribution

摘要: 二项分布是统计问题中常见的分布，对于试验次数n较大时，直接计算的困难很大，因此常用泊松分布或正态分布近似进行计算。本文主要研究二项分布的正态近似，依据中心极限定理，当试验次数n足够大时，可以把服从二项分布的独立同分布的随机变量之和当作正态变量，从而利用正态分布对二项分布近似计算。理论上，二项分布的正态近似只要求n充分大即可，但实际应用中，在n相对较大时，参数p及随机变量取值k的不同对近似计算准确性的影响较为明显。本文用实验法，通过对比独立重复的二项分布与正态近似概率的相对误差得知，对于不同参数p，n值充分大的程度不同。为了更好地得到近似的结果，应根据p的取值，确定n的最小下限值，同时，使用正态近似连续性修正公式可以进一步提高计算的准确性。相较于其他学者的研究，针对不同参数条件的二项分布，本文给出了确定n的最小下限值的方法，一定程度上减少了资源的浪费，为不同试验中n的“充分大”划出了较为明确的定义。

Abstract: Binomial distribution is a common distribution in statistical problems. When the number of tests is large, it is very difficult to calculate directly. Therefore, Poisson distribution or normal distribution approximation is often used for calculation. This paper mainly studies the normal approximation of the binomial distribution. According to the central limit theorem, when the number of tests is large enough, the sum of the random variables that obey the binomial distribution and are independent and identically distributed can be regarded as the normal variable, so that the binomial distribution can be approximated by the normal distribution. Theoretically, the normal approximation of binomial distribution can only be obtained if n is sufficiently large. However, in practical application, when n is relatively large, the difference in the value of parameter p and random variable k has obvious influence on the accuracy of approximate calculation. By comparing the relative error between the binomial distribution of independent repetition and the normal approximation probability, we can know that the minimum lower limit value of n should be de-termined according to the value of p for different parameters with different degrees of sufficient magnitude, so as to obtain better approximation results. At the same time, the accuracy of calcula-tion can be further improved by using the continuity correction formula of normal approximation. Compared with the research of other scholars, this paper gives a method to determine the mini-mum lower limit of n for the binomial distribution under different parameter conditions, which reduces the waste of resources to a certain extent, and draws a more clear definition for the “sufficiently large” of n in different experiments.