概率论毕业论文

概率论毕业 盐城师范学院毕业论文 ;引 言 概率论是一门与现实生活紧密相连的学科，不过大多数人对这门学科的理解还是很 0.50.5平凡的:投一枚硬币，的概率正面朝上，的概率反面朝上，这就是概率论嘛(学过概率论的人又多以为这门课较为理论化，特别是像母函数，极限定理等内容与现实脱节很大，专业性很强(其实如果我们用概率论的对日常生活中的一些看起来比较平凡的内容做些分析，常常会得到深刻的结果( 在谈及应用之前，先澄清一下多数人在概率方面的一个误解(大部分人认为一件事 0概率为，即为不可能事件(这是不对的，比如甲乙玩一个游戏，甲随机地写出一个大01于小于的数，乙来猜(?乙一次猜中这个数?乙每秒猜一次，一直猜下去，“最终” 0猜中这个数(这两件事发生的概率都是，但显然它们都有可能发生，甚至可以“直观”的讲?发生的可能性大些(这说明概率为0的事也是有可能发生的(不过在我看来，这样的可能性实在是太小了，在实际的操作中认为不可能也是有道理的，但不管怎么说，它们确是可能事件( 来看一个应用:[1]在12只金属球中，混有一只假球，并且不知道它是比真球重或轻，用没有砝码的天平来称这些球，试问至少需要多少次称量才能找出这个假球，并确定它是比真球轻或重为了讲清概率论在这个问题中的应用，先讲一下熵的概念(熵是概率论的分支学科--信息论中的概念，它是一个实验不确定程度的量度，熵越大，说明该实验的不确定性越高(比方说，扔一枚硬币是一个实验，扔一枚色子也是一个实验，直观地讲，我们说前者的不确定性要小些;计算结果，前者的熵为，后者的熵为，lg2lg6与直观吻合(同样，判断12个球的真假和轻重也是一个实验，它的熵为，我们要在lg24若干次称量后将其不确定性降为0，也就是要其熵降为0(每用天平称量一次(随便怎样 3k称)，天平都有种结果，于是最多获得的信息，所以次称量最多可得,也lg3k，lg3 kkk,23k,3就是的信息(令得，至少进行次实验才能完成要求(当lg3lg3lg24lg3,, 然，这是理论上最少的结果，我们还要找到一个现实可行的方案，实际上，这样的方案也是有的，所以说得到的解是正确的结果(这种方法将看似是智力测验的题目用方 4法解决了(其实用这种方法还可解决次使用天平，能判断最多多少个球的真假轻重情况的问题(关于这点，可以这样考虑:第一次称量时，所有的球只有两种可能:要么在天平上，要么没有在天平上，且在天平上的球数须是偶数，否则进行的称量是得不到有 2uv用的信息的(设在天平上的球数为，不在天平上的球数为，若天平平衡，下面要3次使用天平在个球中找到假球并判其轻重，由前面的结果知的最大值为12;若天平不平， 2u不妨设其左倾，则假球在个球中，且其轻重已知(若假球是左盘上的一只则假球比spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the ver tical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws第 1 页 共 8 页 2u真球重，否则比真球轻)(判断这个球中哪个球为假球(轻重已判)的实验的熵为，lg2u 2338令，得的最大值是13，于是4次使用天平，最多可判断枚球的真假ulg3lg2lg3,,u 及轻重情况，具体办法也是有的，由于比较繁琐，这里就不列举了(实际上，把这种方 kk(35),法通过观察、归纳、总结，可得更一般的结论:次使用天平多能判断个 (4)k,2 球的真假和轻重状况(这也说明数学的威力所在:它可以将某些东西系统化，得到更一般的结论( 说了这么多，其实就是一个意思，课本上学习的是理论，我们还要尽可能与实际生活联系起来，不要把数学学死了，总之一句话，我们学习数学，是为了更好的认识世界(数学文化，也就是数学在生活中的反映吧(而概率论作为数学的一个分支，与我们的现实生活已是密不可分，了解其发展简史并把概率论作为一个工具应用于生活已是一种必要的修养( 1 概率论的发展简史 概率论同其他数学分支一样，是在一定的社会条件下，通过人类的社会实践和生产活动发展起来的一种智力积累(今日的概率论被广泛应用于各个领域，已成为一棵参天 [2]正如钟开莱1974年所说:“在过去半个世纪中，概率大树，枝多叶茂，硕果累累( 论从一个较小的、孤立的课题发展为一个与数学许多其它分支相互影响、内容宽广而深入的学科(”概率论发展的每一步都凝结着数学家们的心血，正是一代又一代数学家的辛勤努力才有了概率论的今天( 1(1早期的概率现象 人类认识到随机现象的存在是很早的(从太古时代起，估计各种可能性就一直是人类的一件要事(早在古希腊哲学家就已经注意到必然性与偶然性问题;我国春秋时期也已有可考词语(辞海);即使提到数学家记事日程上的可考记载，也至少可推到中世纪(有史记载15世纪上半叶，就已有数学家在考虑这类问题了(如在意大利数学家帕乔利1494 6年出版的《算术》一中就有以下问题:两人进行赌博，规定谁先获胜场谁为胜者(一 2次，当甲已获胜5场，乙也获胜场时，比赛因故中断(那么，赌注该如何分配呢,所 5:2给答案为将赌注分成7份，按分给甲乙两人(当卡丹看到上述问题时，以为所给分 10:1法不妥(他考虑到接下去比赛的几种可能结果，并确定赌注应按来分配(现在看来，其分法也是错误的)(卡丹著有《论赌博》一书，其中提出一些概率计算问题(如掷两spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 盐城师范学院毕业论文 颗骰子出现的点数和的各种可能性等(此外，卡丹与塔塔利亚还考虑了人口统计、保险业等问题(但是他们的研究工作，对数学家来说，赌博味道太浓了一些，以致数学家们对其嗤之以鼻(近代自然科学创始人之一—伽利略解决了以下问题:同时投下三颗骰 6子，点数和为9的情形有种:、、、、和(点(126)，，(135)，，(144)，，(225)，，(234)，，(333)，， 6数和为10的情形也有种:、、、、和，那么(136)，，(145)，，(226)，，(235)，，(244)，，(334)，， 910出现点数和为9与10的机会应相同，而经验告知，出现的机会比出现的机会要多， 925原因何在,伽利略利用列举法得出同时掷三颗骰子出现点数和为的情形有种，而出 1027现点数和为的情形却有种(可见，已经产生了概率论的某些萌芽( 1654年7月29日，法国骑士梅累向数学神童—帕斯卡提出了一个使他苦恼很久的 S问题:“两个赌徒相约若干局，谁先赢了局则赢(若一人赢1局，另一人赢5局，赌 博中止，问赌本应怎么分,”帕斯卡对此思考良久，又将其转给业余数学王子—费马(在数学史上有名的来往信件中，两人取得了一致意见:在被迫停止的赌博中，应当按每个局中人赌赢的数学期望来分配桌面上的赌注(帕斯卡与费马用各自不同的方法解决这个问题，帕斯卡长于计算，运用数学归纳法，推导出数学内含的规律性，而费马以敏锐的观察力，严格的推理，建立起数学概念( 以掷骰子为例来说明他们的解法(即谁先胜3局，则可得到全部赌注，在甲胜2局，乙胜1局时，赌局中止了，问怎样分配赌注才算公平合理( 2帕斯卡分析认为:甲已胜局，乙也胜1局，如再赌一局，则或者甲大获全胜，赢得全部赌金，或者乙胜，则甲与乙胜的局数变成相等，甲、乙应平分赌金(把这两种情 31况平均一下，甲应得赌金的，乙则得赌金的( 44 b费马认为:由甲已胜局，乙已胜局，要结束这场赌博最多还需要赌几局，在这a 个例子中，最多还需要玩两局，结果有四种等可能的情况:(甲胜，甲胜)，(甲胜，乙胜)，(乙胜，甲胜)，(乙胜，乙胜)(在前面三种情况下，甲赢得全部赌金，仅第四种 31情况能使乙获得全部赌金(因此甲有权分得赌金的，而乙应分赌金的( 44 费马和帕斯卡虽然没有明确定义概率的概念，但是，他们定义了使某赌徒取胜的机遇，也就是赢的情况数与所有可能情况数的比，这实际上就是概率，所以概率的发展被认为是从帕斯卡和费马开始的(正如对概率论有卓越贡献的法国数学家泊松后来所说:“由一位广有交游的人向一位严肃的冉森派所提出的一个关于机会游戏的问题乃是概率演算的起源”( spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the ver tical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws第 3 页 共 8 页 当荷兰数学家惠更斯到巴黎的时候，听说帕斯卡与费马在研究概率问题，便也参与进来，并于1657年出版了《论赌博中的计算》一书(书中给出了第一批概率论概念和定理(如加法定理、乘法定理)(在概率论的现代表述中，概率是基本概念，数学期望则是第二级的概念，但在历史上，顺序却相反，先有“期望”概念，而古典概型的概率定义，完全可以从期望概念中导出来(因此，可以认为概率论从此诞生了([3] 1(2成熟中的概率论 最早对概率论来严格化进行尝试的，是俄国数学家伯恩斯坦和奥地利数学家冯?米西斯(他们都提出了一些公理来作为概率论的前提，但他们的公理理论都是不完善的( 作为测度论的奠基人，博雷尔在1905年指出概率论理论如果采用测度论术语来表 1909述将会方便许多，并首先将测度论方法引入概率论重要问题的研究，特别是年他提出并在特殊情形下解决了随机变量序列，服从强大数定律的条件问题(博雷尔的工作 激起了数学家们沿这一崭新方向的一系列探索，其中尤以原苏联数学家科尔莫戈罗夫的研究最为卓著( 从二十世纪二十年代中期起，科尔莫戈罗夫开始从测度论途径探讨整个概率论理论的严格表述(1926年，他推导了弱大数定律成立的主要条件，后又对博雷尔提出的强大数定律问题给出了一般的结果，推广了切比雪夫不等式，提出了科尔莫戈罗夫不等式，创立了可数集马尔可夫链理论，他最著名的工作是1933年以德文出版的经典性著作《概率论基础》(科尔莫戈罗夫是莫斯科函数论学派领导人鲁金的学生，对实际函数论的运用可以说是炉火纯青(他在这部著作中建立起集合测度与事件概率的类比、积分与数学期望的类比、函数正交性与随机变量独立性的类比„„，等等(这种广泛的类比终于赋予了概率论以演绎数学的特征(科尔莫戈罗夫的公理系统逐渐获得了数学家们的普遍承认，由于公理化，概率论成为一门严格的演绎科学，取得了与其他数学分支同等的地位( 科尔莫戈罗夫热爱教育事业，经常在大学生和进修生中挑选人才，参加讨论班(1934年，他与概率论另一位创始人辛钦共同主持概率论讨论班(在他们培养的学生中有6位成为前苏联科学院院士或通信院士(1980年科尔莫戈罗夫荣获沃尔夫奖([4] 公理化概率论首先使随机过程的研究获得了新的起点，随机过程作为随时间变化的偶然量的数学模型，是现代概率论研究的重要主题( 莱维从1938年开始创立研究随机过程的新方法，即着眼于轨道性质的概率方法( 1948年出版的《随机过程与布朗运动》，提出了独立增量过程的一般理论，并以其为基础极大地推进了对作为一类特殊马尔可夫过程的布朗运动的研究(1939年维尔引进“鞅”这个名称，但鞅论的奠基人是美国概率论学派的代表人物杜布(杜布从1950年spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 盐城师范学院毕业论文 开始对鞅概念进行了系统的研究而使鞅论成为一门独立的分支(鞅论使随机过程的研究进一步抽象化,不仅丰富了概率论的内容,而且为其他数学分支如调和分析、复变函数、位势理论等提供了有力的工具(从1942年开始,日本数学家伊藤清引进了随机积分与随机微分方程,为一门意义深远的数学新分支——随机分析的创立与发展奠定了基础([5] 概率论不仅是“数学之树”的一庞大支条，而且还有若干强壮的根(如下表)，直接扎在实际应用环境的大地上(“芳草有情皆碍马，好云无处不遮楼”(正如英国的逻辑学家和经济学家杰文斯所说，概率论是“生活真正的领路人，如果没有对概率的某种估计，我们就寸步难行，无所作为(” 2 概率统计在实际生活中的应用 2(1关于男女色盲比例的问题 8名色盲，而433名女性中发现1人色盲，例1[6]从随机抽取的467名男性中发现有 ,,0.01在水平上能否认为女性色盲的比例比男性低, 解 设男性色盲的比例为,女性色盲的比例为，那么要检验的假设为 pp12 H:p,pH:p,p012112 ,,0.01由备择假设，利用大样本的正态近似得，在水平的拒绝域为 ,,u,,2.33 由样本得到的结果知: n,467,m,433 818，1ˆˆˆp,,0.01713,p,,0.00231,p,,0.1 12467433467，433 则 ˆˆp,p12u,,2.2326 11,,ˆˆ，，，p1,p,,nm,, ,,0.01未落在拒绝域中，因此在水平上可以认为女性色盲的比例低于男性( 2(2我国出生人口性别比 出生人口性别比，通常是为了便于观察与比较所定义的每出生百名女婴相对的出生男婴数(20世纪50年代中期，联合国在其出版的《用于总体估计的基本数据质量鉴定 100方法》(手册?)认为:出生性别比偏向于男性(一般来说，每出生名女婴，其男spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the ver tical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws第 5 页 共 8 页 102,107102,107婴出生数置于之间(此分析明确认定了出生性别比的通常值域为之 间(从此出生性别比值下限不低于102、上限不超过107的值域一直被国际社会公认为 通常理论值，其他值域则被视为异常( 例2近年来，越来越多的话题围绕着我国的人口性别比例而展开(下图(表1)所 示的是我国2005年到2010年的出生人口性别比例的变化情况( 由图可以看出，在2005年到2010年之间，我国的人口性别比一直都保持在118到 102107,121之间，超出了国际社会公认为通常理论值很多( 3(3电影院的座位问题 2x,R定理1 设，则对任意,有 DX,,i2005-2010年中国人口性别比2u,x,,1X,a2,,lim，，P,x,edu,,x ,,,,,,,n,2,n,, Xa,~N，，0,1.记为 ,n 这一结果称为Lindeberg-Levy定理，是这两位学者在20世纪20年代证明的(历 Laplace 定理，它是前一个结果史上最早的中心极限定理是1716年建立的De Moivre- 的特例，具体为 nXnp,lim()()([7] pxx,,,x,,(1)npp, 3n,1600例3设某地扩建电影院，据分析平均每场观众数人，预计扩建后，平均的 4 观众仍然会去该电影院，在设计座位时，要求座位数尽可能多，但空座达到200或更多 0.1的概率不能超过，问应该设多少座位, spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws 122 120.56 121 120.22 119.45120 119.25 118.58 119 118.06 118 117 116 200520062007200820092010 盐城师范学院毕业论文 解 把每日看电影的人编号为，且令 1,2,？,1600 1，第个观众还去电影院i,( Xi,,121600，，？,i0，不然, 31则由题意(又假定各观众去电影院是独立选择，则pXpX,,,,，(1)(0)ii44 是独立随机变量，现设座位数为，则按要求 mX,X,？12 ( pXXXm(2000.1，，，,,,？)121600 3在这个条件下取最大(当上式取等号时，取最大，因为，np,，,16001200mm4 ，由定理第二个式子知，应满足 mnpp(1)103,, m,,2001200( ,,()0.1 103 m,1377查正态分布表即可确定，所以，应该设1377个座位( 3 总结 兴趣是最好的老师，可以激发学生的学习热情，更可以引导学生成为学习的主人，学习数学需要死记硬背熟能生巧，但并不排除用兴趣引导和激励(将兴趣转化为志趣，转化为学习的动力，将其带到数学学习的每一个部分(本文我们主要通过讲解三个生活中遇到的悖论问题，使人们在生活与学习中，能更好的理解悖论给我们带来的困惑，解决了人们在意识上的一些错误观点(对于这些因为意识的错觉而存在的悖论问题，我们仍有待于进一步研究(上面列举了概率统计在实际生活中的一些简单应用，其实日常生活中到处都有概率统计的影子(通过统计我们可以了解一些指数的变化趋势等，通过概率计算我们了解了彩票、摸奖等的中奖率等(概率统计的足迹可以说是已经深入到每一个领域，在实际问题的应用随处可见(相信人类能够更好的应用好概率统计，使之更好的为人类的发展做贡献( spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the ver tical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws第 7 页 共 8 页 参考文献 [1]梅长林,周家良.实用统计方法[M].北京:科学出版社,2002. [2]杨虎,钟波,刘琼荪.应用数理统计[M].北京:清华大学出版社,2006. [3]张国权.应用概率统计[M]. 北京:科学出版社,2003. [4]吴传志.应用概率统计[M].重庆:重庆大学出版社,2004. [5]郑长波.生活中的概率问题举例[J].沈阳师范大学学报,2007,7(5):23-26. [6]魏宗舒,等.概率论与数理统计教程[M].北京:高等教育出版社,2008. [7]王梓坤.概率论基础及其应用[M].北京:科学出版社,1976. spacing of 100mm. C, line line line nearby. D, spare wire unified terminal strip Terminal (terminal strip at the top of the vertical, horizontal terminal strip from the harness to Terminal Pai end at the beginning), length to ensure as far as Terminal and set aside. After e, form a wire harness, cable, line. When the line should be in the core wire for better identification and marking on the cards will be prepared in advance, alternate core core content for the cable numbering. F, finish line, according to the corresponding terminal locations, wire one by one from the bunch out, pulling line, should be kept parallel to each other and more. Core dark as possible, determined in accordance with specific conditions, it is required that: neat, beautiful, shapely, pleasing to the eye. 7.6.2 wiring technology, out of the core wire according to Terminal position, cut off the excess, with wire stripping pliers or electrician knife stripped of insulation, so that connection. Stripping should not damage the copper core. B, core wire on the oxide and chip-insulated electricians knife scrape. C, core processing has been completed, thread round long nose pliers in a clockwise direction. Circle with the screw in the direction of spin in the same direction, circle should be appropriate, it is best to slightly larger than the screws, bent round the circle and the length of root and appropriate so set screw is screwed on tight. D, when on the crimp-type terminals when wiring, threads do not have to bend. Insulating part should be careful not to press it into a Terminal, and make the loop doesn't make sense and to prevent bad head crimping, fell down, causing an open circuit. After wiring, may be tried. E, crimping Terminal crimping, wire and Terminal contact is good, solid, not subject to mechanical strain, and ensure the insulation level. F, cables, such as when there is a bare copper ground wire, bare copper wire with a plastic pipe should be set. G, when wiring wiring must be in radians. H, terminal screws