《概率与数理统计》复习材料 - 图文

《概率论与数理统计》复习材料

集美大学诚毅学院 201241073056编 2014.06.02

目 录

一、随机事件 ······································································································································································ - 3 -

1、一些术语 ································································································································································ - 3 - 2、事件间的关系 ························································································································································ - 3 - 3、事件间的运算规律 ················································································································································ - 4 - 4、易混淆的语句 ························································································································································ - 4 - 5、例题 ········································································································································································ - 4 - 二、概率 ·············································································································································································· - 4 -

1、高中基础 之 阶乘、排列、组合 ························································································································ - 4 - 2、概率与频率的区别 ················································································································································ - 5 - 3、概率的性质 ···························································································································································· - 5 - 4、古典概型 ································································································································································ - 5 - 5、加法法则、条件概率、乘法法则 ························································································································ - 5 - 6、一些公式 ································································································································································ - 5 - 7、全概率定理与贝叶斯定理 ···································································································································· - 6 - 8、独立试验概型 ························································································································································ - 6 - 9、贝努里定理 ···························································································································································· - 6 - 10、例题······································································································································································ - 6 - 三、一元随机变量 ···························································································································································· - 13 -

1、表示方法：

?、?、?、X、Y、Z ··············································································································· - 13 -

2、“离散型随机变量”与非离散型随机变量中的“连续型随机变量” ··························································· - 13 - 3、例题 ······································································································································································ - 14 - 四、二元随机变量 ···························································································································································· - 17 -

1、离散型与连续型 ·················································································································································· - 17 - 2、例题 ······································································································································································ - 19 - 五、随机变量的数字特征 ················································································································································ - 25 -

1、数学期望（是确定的一个数，不是变量） ······································································································ - 25 - 2、方差 ······································································································································································ - 26 - 3、标准差 ·································································································································································· - 26 - 4、协方差 ·································································································································································· - 26 - 5、例题 ······································································································································································ - 26 - 六、几种重要的分布 ························································································································································ - 31 -

1、离散型 ·································································································································································· - 31 - 2、连续型 ·································································································································································· - 32 - 3、例题 ······································································································································································ - 34 -

- 1 -

七、大数定律与中心极限定理 ········································································································································ - 40 -

1、大数定律 ······························································································································································ - 40 - 2、中心极限定理 ······················································································································································ - 40 - 3、例题 ······································································································································································ - 42 - 八、样本分布 ···································································································································································· - 46 -

1、一些术语 ······························································································································································ - 46 - 2、样本平均数和样本方差的简算公式 ·················································································································· - 47 - 3、几个常用统计量的分布 ······································································································································ - 47 - 4、例题 ······································································································································································ - 48 - 九、参数估计 ···································································································································································· - 50 -

1、估计量 ·································································································································································· - 50 - 2、评价估计量好坏的三种最常用的标准 ·············································································································· - 50 - 3、获得估计量的方法 ·············································································································································· - 51 - 4、例题 ······································································································································································ - 53 - 十、假设检验 ···································································································································································· - 59 -

1、一些术语 ······························································································································································ - 59 - 2、用置信区间的方法进行检验的基本思想（方便理解）··················································································· - 59 - 3、两类错误 ······························································································································································ - 59 - 4、一个正态总体的假设检验 ·································································································································· - 59 - 5、例题 ······································································································································································ - 61 -

- 2 -

一、随机事件

1、一些术语 基本事件 必然事件 不可能事件 样本空间 样本点

不能分解成其它事件组合的最简单的随机事件 符号为Ω 符号为Φ 随机试验所有可能的结果组成的集合称为样本空间，记为Ω 随机试验的每个可能结果，记为? 2、事件间的关系 用A?B（事件A含于事件B） 或 B?A（事件包含 B包含事件A）表示 对于任何事件A，有Φ?A?Ω 相等 并 （和） 用A?B表示 若A?B且B?A，则A?B 用A?B 或 A?B表示 即“A和B至少有一个发生”,“A发生或B发生” 交 （积） 用AB 或 A?B表示 即“A发生且B发生” 差 互斥事件 (互不相容事件) 对立事件 用A ?B表示 即“A发生且B不发生” A ?B?AB ?A?AB 用AB ?Φ表示 若AB ?Φ，则称A、B相容 若AB?Φ且A?B?Ω，则称AB互为对立事件, 记作A?B,B?A 若事件A1,A2,?,An为两两互不相容的事件，并且 完备事件组 A1?A2???An?Ω，则称A1,A2,?,An构成一个完备事件组

- 3 -

3、事件间的运算规律

注：A?B可简写为AB 交换律 结合律 分配律 德·摩根律 （对偶律）

A?B?B?A，AB?BA (A?B)?C?A?(B?C) ， (AB)C?A(BC)(A?B) C?AC?BC (AB) ?C?(A ?C)(B ?C) A?B?A B ， AB?A?B 4、易混淆的语句

“都没有”与“没有都”。买三次彩票,“三次都没中”与“没有三次都中”是不一样的

5、例题

1 PPT 2 PPT 利用事件关系和运算 表达以下事件的关系：(1)A ,B ,C 都不发生；(2)A ,B ,C 不都发生 【答案】(1)A B C（或A?B?C ）. (2)ABC （或A?B?C） 在图书馆中随意抽取一本书，事件A表示数学书，B表示中文书，C表示平装书，用文字叙述下列事件：①ABC，②C?B 【答案】(1)抽取的是精装中文版数学书. (2)精装书都是中文书 一批产品有合格品和废品，从中有放回地抽取三个产品，设A1,A2,A3分别表示第1,2,3次抽到废品，(1)请用文字叙述下列事件A?A1?A2?A3: ; 3 作业 B?A1 A2 A3: ; C?A1A2A3: . (2)A、B、C中 和 为对立事件. (3)请用A1,A2,A3的运算关系式表示下列事件：第一次抽到合格品: ； 只有第一次抽到合格品 ；只有一次抽到合格品 . 【答案】(1)三次至少有一次抽到废品；三次都没有抽到废品；三次至少有一次抽到合格品. (2)A和B. (3)A1 ; A1 A2 A3 ; A1 A2 A3?A1 A2 A3?A1 A2 A3

二、概率

1、高中基础 之 阶乘、排列、组合

阶乘 排列（有序） 组合（无序）

①N！=1×2×…×N（N为正整数） ②0 ! = 1 ①nm?n! An?n(n?1)(n?2)?(n?m?1) ②AnmnmmAnC ②?nm!①Cnn?m?Cn ③Cn?1 - 4 -

2、概率与频率的区别 频率 在n次重复试验中，若事件A发生了m次，则m称为事件A发生的频率 n在条件不变的情况下，重复进行n次试验，事件A发生的频率稳定地在某一A的概率，记作P(A)?p 概率 常数p附近摆动。且一般说来，n越大，摆动幅度就越小，则称常数p为事件

3、概率的性质

0≤P≤1，P(Ω)?1 , P(Φ)?0

注：不可能事件的概率为0，但概率为0的事件不一定是不可能事件； 必然事件的概率为1，但概率为1的事件不一定是必然事件。

4、古典概型

?1,?2,?,?n?（有限个样本点），且P(?1)?P(?2)???P(?n)， 设试验中Ω?? 则有P(A)?

A的有利样本点集所含样本点数

Ω样本点数 5、加法法则、条件概率、乘法法则

(1) 加法法则：当事件A、B互斥（即互不相容）时，P(A+B)=P(A)+P(B)

P(AB)（B是条件）（称作“A对B的条件概率”） P(B)P(AB) ②P(B|A)?（A是条件）（称作“B对A的条件概率”）

P(A)P(B|A)?P(A) · (3) 乘法法则：P(AB)??

P(B) ·P(A|B)? (2) 条件概率：①P(A|B)?P(C|AB)?P(A) ·P(B|A) ·P(C|AB), 注：将AB视为整体，可推出P(ABC)?P(AB) · 依此类推可得P(A1 A2…An) = P(A1)·P (A2 | A1)·P(A3 | A1 A2) … P(An | A1 A2 …An-1 )

6、一些公式

(1) 概率的可列可加性：若事件A1,A2,?,An两两不相容，则 P(A1?A2???An)?P(A1)?P(A2)???P(An) (2) 完备事件组的概率：P(A1?A2???An)?1 (3) 对立事件概率和为1：P(A)+P(A)=1，P(A)=1-P(A) (4) P(B-A) = P(B)-P(AB) （草稿画图）

(5) 当A?B（或B?A）时，显然P(A)≤P(B)，此时有P(B-A) = P(B)-P(A) (6) P(A+B) = P(A)+P(B)-P(AB)≤P(A)+P(B) （草稿画图）

(7) P(A+B+C) = P(A) + P(B) + P(C) + P(ABC)-P(AB)-P(AC)-P(BC)

- 5 -

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说综合文库《概率与数理统计》复习材料 - 图文在线全文阅读。