由於電腦發展的普及，在許多的科學研究裡，電腦經被當作輔助的工具。在統計的領域裡，常會參考電腦模擬的結果，來處理較為複雜的問題；而在數學方面，也可利用電腦模擬的方法，來解決數值分析上的問題，例如蒙地卡羅應用在多維度的積分上。所以，如何產生良好的隨機數數列是值得我們去深入研究的課題。然而，一個好的隨機數產生器(Random Number Generator；RNG)必須滿足下列之特性：1. 隨機性(Randomness)、2. 長週期(Long Period)、3. 有效性(Efficiency)、4. 可攜帶性(Portability)、5. 可重複性(Repeatability)。
本文主要是利用Visual Basic 6.0來撰寫隨機數產生器及統計檢定之程式。在隨機數產生器方面，選擇較被廣泛使用的線性同餘產生器(Linear Congruential Generator；LCG），具有目前最長週期紀錄的DX-1511 法(多項遞迴法的推廣)以及組合法。對隨機數產生器所產生之隨機數列做各項比較統計及檢定，得到了DX1511法具有相對較好的統計性質，故針對DX-1511法改變遞迴式中的係數及modulus p，以驗證其穩定性。同時利用二維格子結構圖以比較各隨機數產生器，在兩維度的平面是否具有某些規律性存在。本文主要是透過多種比較統計檢定方法能夠分辨出各隨機數產生器之差異。

Because the efficiency and less cost of computer, the computer is used to contribute many scientific researches. In recent researches, we often deal with complicate problems by referring to the results of the computer simulations. We can also numerically solve problems, such as doing numerical analysis and applying Monte Carlo method for high dimensional integrals. As a consequence, the subject, how to generate better random number sequences, is worthy to investigate. A fine random number sequence has to satisfy the following five characteristics: 1.Randomness 2.Long Period 3.Efficiency 4.Portability 5.Repeatability.
In this thesis, I distinguish the differences of random number generators by ten statistical tests. I write programs of random number generators and statistical tests mainly by using software Visual Basic 6.0 and Maple 9. For random number generators, I choose the Linear Congruential Generator (LCG) which is widely used in recent times, the DX-1511 which is a kind of Multiple Recursive Generator (MRG) with the longest period in current record, and the combination generator. I compare statistical properties by using various empirical tests on random sequences generated from these random number generators. I obtain the result that the DX-1511 generator is the champion considering all the five mentioned characteristics. As a consequence, I also demonstrate the stability of the DX-1511 Generator by changing the multiplier coefficient a and the modulus p. I also judge against these random number generators by using two-dimension lattice plot, and evaluate if there exist some patterns.

目錄
第一章 緒論………….………………………….……………1
第一節 研究背景與動機……………………………………1
第二節 研究目的……………………………………………3
第三節 研究步驟……………………………………………3
第四節 研究範圍與限制............................5
第二章 文獻探討…………...…………………………………6
第三章 研究方法…………...………………………………..14
第一節 統計檢定……………………………………….….14
第二節 隨機數產生器之基本性質…………………….….20
第四章 資料分析…………………………………………….21
第一節 隨機數產生器的選取……………………………..21
第二節 隨機數之統計檢定………………………………..23
第三節 隨機數產生器之評比……………………………..54
第四節 改變DX-1511係數的比較…………………….…58
第五節 二維格子結構圖……………………………….….80
第五章 結論與建議………………………………………….85
第一節 結論…………………………………………….…85
第二節 建議…………………………………………….…88
參考文獻…………………...………………………………....89
附錄一(visual Basic 6.0隨機數產生器程式碼)…………….94
附錄二(Visual Basic 6.0統計檢定程式碼)…………............96
附錄三(Maple 9隨機數產生器程式碼)…………………...105
附錄四(Maple 9二維格子圖程式碼)……………………...106
圖目錄
圖1.3.1：研究架構圖………………………………………………….4
圖4.5.1：二維格子結構圖比較(X軸之值固定在(0,1))..…………...81
圖4.5.2：二維格子結構圖比較(X軸之值固定在(0.70,0.71)).….….82
圖4.5.3：二維格子結構圖比較(X軸之值固定在(0.700,0.701))…...83
圖4.5.4：二維格子結構圖比較(X軸之值固定在(0.7000,0.7001))...84
表目錄
表 4.2.1：卡方檢定例子……………………………………………..23
表4.2.2：均等區間的卡方檢定模擬結果…………………………...24
表4.2.3：升降趨勢檢定模擬結果…………………………………...26
表4.2.4：樸克檢定例子……………………………………………...27
表4.2.5：樸克檢定模擬結果………………………………………...28
表4.2.6：二項分配的卡方檢定例子(n=10)…………………………29
表4.2.7：二項分配的卡方檢定模擬結果(n=10)……………………30
表4.2.8：二項分配的卡方檢定模擬結果(n=20)……………………31
表4.2.9：二項分配的卡方檢定模擬結果(n=30)……………………32
表4.2.10：二項分配的卡方檢定模擬結果(n=40)…………………..33
表4.2.11：二項分配的卡方檢定模擬結果(n=50)…………………..34
表4.2.12：和的常態檢定例子(n=36)………………………………..35
表4.2.13：和的常態檢定模擬結果(n=36)…………………………..36
表4.2.14：和的常態檢定模擬結果(n=64)…………………………..37
表4.2.15：和的常態檢定模擬結果(n=100)…………………………38
表4.2.16：序列檢定例子(d=5)………………………………………39
表4.2.17：序列檢定模擬結果(d=5)…………………………………40
表4.2.18：序列檢定模擬結果(d=10)………………………………..41
表4.2.19：排列檢定例子…………………………………………….42
表4.2.20：排列檢定模擬結果……………………………………….43
表4.2.21：間隔檢定例子…………………………………………….44
表4.2.22：間隔檢定模擬結果……………………………………….45
表4.2.23：K-S檢定例子……………………………………………..46
表4.2.24：K-S檢定模擬結果………………………………………..47
表4.2.25：相關檢定模擬結果(a=2)………………………………….49
表4.2.26：相關檢定模擬結果(a=11)………………………………...50
表4.2.27：相關檢定模擬結果(a=73)………………………………...51
表4.2.28：相關檢定模擬結果(a=3)………………………………….52
表4.2.29：隨機數產生器統計檢定整理表…………………………..53
表4.3.1：電腦執行時間……………………………………………....55
表4.3.2：利用VB 6.0與Maple 9產生LCG法隨機數列………….55
表4.3.3：利用VB 6.0與Maple 9產生DX-1511-2法隨機數列…...56
表4.3.4：利用VB 6.0與Maple 9產生DX-1511-2法隨機數列…...56
表4.3.5：利用VB 6.0與Maple 9產生組合法隨機數列…………...56
表4.3.6：隨機數產生器基本性質整理表……………………………57
表4.4.1：改變DX-1511法係數均等區間的卡方檢定模擬結果…...59
表4.4.2：改變DX-1511法係數升降趨勢檢定模擬結果…………...60
表4.4.3：改變DX-1511法係數樸克檢定模擬結果………………...61
表4.4.4：改變DX-1511法係數二項分配的卡方檢定模擬結果(n=10).62
表4.4.5：改變DX-1511法係數二項分配的卡方檢定模擬結果(n=20).63
表4.4.6：改變DX-1511法係數二項分配的卡方檢定模擬結果(n=30).64
表4.4.7：改變DX-1511法係數二項分配的卡方檢定模擬結果(n=40).65
表4.4.8：改變DX-1511法係數二項分配的卡方檢定模擬結果(n=50).66
表4.4.9：改變DX-1511法係數和的常態檢定模擬結果(n=36)……67
表4.4.10：改變DX-1511法係數和的常態檢定模擬結果(n=64)…..68
表4.4.11：改變DX-1511法係數和的常態檢定模擬結果(n=100)…69
表4.4.12：改變DX-1511法係數序列檢定模擬結果(d=5)…………70
表4.4.13：改變DX-1511法係數序列檢定模擬結果(d=10)………..71
表4.4.14：改變DX-1511法係數排列檢定模擬結果……………….72
表4.4.15：改變DX-1511法係數間隔檢定模擬結果……………….73
表4.4.16：改變DX-1511法係數K-S檢定模擬結果……………….74
表4.4.17：改變DX-1511法係數相關檢定模擬結果(a=2)………….75
表4.4.18：改變DX-1511法係數相關檢定模擬結果(a=11)………...76
表4.4.19：改變DX-1511法係數相關檢定模擬結果(a=73)………...77
表4.4.20：改變DX-1511法係數相關檢定模擬結果(a=3)………….78
表4.4.21：改變DX-1511係數統計檢定整理表……………………..79

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