- 學位論文
常用機率分配的常態近似性
Normal Approximation of Frequently Used Probability Distributions
摘要
在中央極限定理（Central Limit Theorem）的基礎上，當樣本數大於30 時，其樣本平均數的抽樣分配是近似常態分配，也就是在中央極限定理下，常態分配可當做不少大樣本的近似分配。然在現實生活中存在著各式形態的機率分配；如有單峰、多峰之分配；對稱、不對稱之分配；高、低偏態之分配；以及無偏、無峰、無尾且對稱之均勻分配，各個機率分配之形態有的與常態分配類似，有的則差距很大，甚至有由常態分配所衍生的其他分配，如 t分配、卡方分配、F 分配，其中 t分配形如標準常態分配，卡方分配、F 分配形如Gamma分配，其峰度及偏態隨自由度而變化。本文欲探究樣本數(自由度)應該要多大，t分配、卡方分配及二項分配才可以被接受近似常態分配，而以常態分配所取代，也探討在實際應用中央極限定理時，一般教科書所建議的樣本大小是否合適的問題。 本研究試圖運用電腦查表方式，針對 t分配、卡方分配與標準常態分配臨界值的誤差做比較及使用電腦模擬方式，針對卡方分配及二項分配樣本平均數的抽樣分配近似常態分配所需的最小樣本數(自由度)做探討，並在各章節提供各機率分配之近似常態分配所需的最小樣本數(自由度)之參考表格。
並列摘要
According to the Central Limit Theorem, when the sample n>30 size , the sampling distribution of sample mean will be approximated to the normal distribution. In other words, the normal distribution can be used as the approximate distribution of many types of samples under the Central Limit Theorem. In real life, there are assorted types of probability distribution, such as the unimodal vs multimodal distributions, the symmetrical vs asymmetrical distributions, the high vs low skewness distributions, and the non-skewed, nonmodal, and no-tail uniform distribution. While some of probability distributions have a normal distribution-like pattern, others may have a pattern that differs greatly from the normal distribution pattern. Take the normal distribution derived t-distribution, chi-square distribution, F distribution as an example, even though the t-distribution has a shape like that of the standard normal distribution, and chi-square distribution, such as the gamma distribution, its kurtosis and skewness change according to the degree of freedom, commonly. The purpose of this thesis is to explore the minimum sample size ( degrees of freedom) required of the t-distribution, chi-square distribution and binomial distribution for the normal approximation and be replaced by the normal distribution and to explore the investigators examined the appropriateness of using the sample size suggested by general textbooks for determining whether the Central limit theorem can be used or not. It was done using the computer technology to compare the critical value of the error at the t-distribution,chi-square distribution and Z-distribution and used the computer simulation to explore the minimum sample size ( degrees of freedom) and required for the means of the chi-square distribution and binomial distribution to be approximated to the normal distribution The minimum sample size (degrees of freedom) required for the normal distribution approximating to each probability distribution is also listed in the table in various chapters for reference.