概率论与数理统计复习
第一章 概率论的基本概念
一.基本概念
随机试验E:(1)可以在相同的条件下重复地进行;(2)每次试验的可能结果不止一个,并且能事先明确试验的所有可能结果;(3)进行一次试验之前不能确定哪一个结果会出现. 样本空间S: E 的所有可能结果组成的集合. 样本点(基本事件):E 的每个结果. 随机事件(事件):样本空间S 的子集.
必然事件(S):每次试验中一定发生的事件. 不可能事件(Φ):每次试验中一定不会发生的事件. 二. 事件间的关系和运算
1.A ?B(事件B 包含事件A )事件A 发生必然导致事件B 发生.
2.A ∪B(和事件)事件A 与B 至少有一个发生.
3. A ∩B=AB(积事件)事件A 与B 同时发生.
4. A -B(差事件)事件A 发生而B 不发生.
5. AB=Φ (A 与B 互不相容或互斥)事件A 与B 不能同时发生.
6. AB=Φ且A ∪B=S (A 与B 互为逆事件或对立事件)表示一次试验中A 与B 必有一个且仅有一个发生. B=A, A=B .
运算规则 交换律 结合律 分配律 德?摩根律 B A B A = B A B A = 三. 概率的定义与性质
1.定义 对于E 的每一事件A 赋予一个实数,记为P(A),称为事件A 的概率. (1)非负性 P(A)≥0 ; (2)归一性或规范性 P(S)=1 ;
(3)可列可加性 对于两两互不相容的事件A 1,A 2,…(A i A j =φ, i ≠j, i,j=1,2,…),
P(A 1∪A 2∪…)=P( A 1)+P(A 2)+… 2.性质
(1) P(Φ) = 0 , 注意: A P(A)=0 .
(2)有限可加性 对于n 个两两互不相容的事件A 1,A 2,…,A n ,
P(A 1∪A 2∪…∪A n )=P(A 1)+P(A 2)+…+P(A n ) (有限可加性与可列可加性合称加法定理) (3)若A ?B, 则P(A)≤P(B), P(B -A)=P(B)-P(A) . (4)对于任一事件A, P(A)≤1, P(A)=1-P(A) .
(5)广义加法定理 对于任意二事件A,B ,P(A ∪B)=P(A)+P(B)-P(AB) . 对于任意n 个事件A 1,A 2,…,A n
()()()
()
+∑
+
∑
-
∑=≤<<≤≤<≤=n
k j i k j i n
j i j i n
i i n A A A P A A P A P A A A P 111
21
…+(-1)n-1P(A 1A 2…A n )
四.等可能(古典)概型
1.定义 如果试验E 满足:(1)样本空间的元素只有有限个,即S={e 1,e 2,…,e n };(2)每一个基本事件的概率相等,即P(e 1)=P(e 2)=…= P(e n ).则称试验E 所对应的概率模型为等可能(古典)概型.
2.计算公式 P(A)=k / n 其中k 是A 中包含的基本事件数, n 是S 中包含的基本事件总数. 五.条件概率
1.定义 事件A 发生的条件下事件B 发生的条件概率P(B|A)=P(AB) / P(A) ( P(A)>0).
2.乘法定理 P(AB)=P(A) P (B|A) (P(A)>0); P(AB)=P(B) P (A|B) (P(B)>0).
P(A 1A 2…A n )=P(A 1)P(A 2|A 1)P(A 3|A 1A 2)…P(A n |A 1A 2…A n-1) (n ≥2, P(A 1A 2…A n-1) > 0) 3. B 1,B 2,…,B n 是样本空间S 的一个划分(B i B j =φ,i ≠j,i,j=1,2,…,n, B 1∪B 2∪…∪B n =S) ,则 当P(B i )>0时,有全概率公式 P(A)=()()i n
i i B A P B P ∑=1
当P(A)>0, P(B i )>0时,有贝叶斯公式P (B i |A)=()()()()()()
∑==n i i i i i i B A P B P B A P B P A P AB P 1
.
六.事件的独立性
1.两个事件A,B,满足P(AB) = P(A) P(B)时,称A,B 为相互独立的事件. (1)两个事件A,B 相互独立? P(B)= P (B|A) .
(2)若A 与B ,A 与B ,A 与B, ,A 与B 中有一对相互独立,则另外三对也相互独立. 2.三个事件A,B,C 满足P(AB) =P(A) P(B), P(AC)= P(A) P(C), P(BC)= P(B) P(C),称A,B,C 三事件两两相互独立. 若再满足P(ABC) =P(A) P(B) P(C),则称A,B,C 三事件相互独立. 3.n 个事件A 1,A 2,…,A n ,如果对任意k (1 ()()()() k k i i i i i i A P A P A P A A A P 2 1 2 1 =,则称这n 个事件A 1,A 2,…,A n 相互独立. 第二章 随机变量及其概率分布 一.随机变量及其分布函数 1.在随机试验E 的样本空间S={e}上定义的单值实值函数X=X (e)称为随机变量. 2.随机变量X 的分布函数F(x)=P{X ≤x} , x 是任意实数. 其性质为: (1)0≤F(x)≤1 ,F(-∞)=0,F(∞)=1. (2)F(x)单调不减,即若x 1 1.离散型随机变量的分布律 P{X= x k }= p k (k=1,2,…) 也可以列表表示. 其性质为: (1)非负性 0≤P k ≤1 ; (2)归一性 11=∑∞ =k k p . 2.离散型随机变量的分布函数 F(x)=∑≤x X k k P 为阶梯函数,它在x=x k (k=1,2,…)处具有跳跃点, 其跳跃值为p k =P{X=x k } . 3.三种重要的离散型随机变量的分布 (1)X~(0-1)分布 P{X=1}= p ,P{X=0}=1–p (0 (2)X~b(n,p)参数为n,p 的二项分布P{X=k}=()k n k p p k n --??? ? ??1(k=0,1,2,…,n) (0 λ-e k k ! (k=0,1,2,…) (λ>0) 三.连续型随机变量 1.定义 如果随机变量X 的分布函数F(x)可以表示成某一非负函数f(x)的积分F(x)=()dt t f x ?∞-,-∞< x <∞,则称X 为连续型随机变量,其中f (x)称为X 的概率密度(函数). 2.概率密度的性质 (1)非负性 f(x)≥0 ; (2)归一性 ?∞ ∞-dx x f )(=1 ; (3) P{x 1 1 )(x x dx x f ; (4)若f (x)在点x 处连续,则f (x)=F / (x) . 注意：连续型随机变量X 取任一指定实数值a 的概率为零,即P{X= a}=0 . 3.三种重要的连续型随机变量的分布 (1)X ～U (a,b) 区间(a,b)上的均匀分布 ???=-0 )(1a b x f 其它b x a << . (2)X 服从参数为θ的指数分布.()???=-0 /1θθx e x f 00≤>x x 若若 (θ>0). (3)X~N (μ,σ2 )参数为μ,σ的正态分布 2 22)(21)(σμσ π--= x e x f -∞ 特别, μ=0, σ2 =1时,称X 服从标准正态分布,记为X~N (0,1),其概率密度 2 221 )(x e x -= π ? , 标准正态分布函数 ?= Φ∞--x t dt e x 2221 )(π , Φ(-x)=1-Φ(x) . 若X ～N ((μ,σ2), 则Z= σ μ -X ~N (0,1), P{x 1 σμ-2x )-Φ( σ μ -1x ). 若P{Z>z α}= P{Z<-z α}= P{|Z|>z α/2}= α,则点z α,-z α, ±z α/ 2分别称为标准正态分布的上,下,双侧α分位点. 注意：Φ(z α)=1-α , z 1- α= -z α. 四.随机变量X 的函数Y= g (X)的分布 1.离散型随机变量的函数 若g(x k ) (k=1,2,…)的值全不相等,则由上表立得Y=g(X)的分布律. 若g(x k ) (k=1,2,…)的值有相等的,则应将相等的值的概率相加,才能得到Y=g(X)的分布律. 2.连续型随机变量的函数 若X 的概率密度为f X (x),则求其函数Y=g(X)的概率密度f Y (y)常用两种方法： (1)分布函数法 先求Y 的分布函数F Y (y)=P{Y ≤y}=P{g(X)≤y}=()()dx x f k y X k ∑?? 其中Δk (y)是与g(X)≤y 对应的X 的可能值x 所在的区间(可能不只一个),然后对y 求导即得f Y (y)=F Y /(y) . (2)公式法 若g(x)处处可导,且恒有g /(x)>0 (或g / (x)<0 ),则Y=g (X)是连续型随机变量,其 概率密度为 ()()()() ? ??'=0y h y h f y f X Y 其它βα< 其中h(y)是g(x)的反函数 , α= min (g (-∞),g (∞)) β= max (g (-∞),g (∞)) . 如果f (x)在有限区间[a,b]以外等于零,则 α= min (g (a),g (b)) β= max (g (a),g (b)) . 第三章 二维随机变量及其概率分布 一.二维随机变量与联合分布函数 1.定义 若X 和Y 是定义在样本空间S 上的两个随机变量,则由它们所组成的向量(X,Y)称为二维随机向量或二维随机变量. 对任意实数x,y,二元函数F(x,y)=P{X ≤x,Y ≤y}称为(X,Y)的(X 和Y 的联合)分布函数. 2.分布函数的性质 (1)F(x,y)分别关于x 和y 单调不减. (2)0≤F(x,y)≤1 , F(x,- ∞)=0, F(-∞,y)=0, F(-∞,-∞)=0, F(∞,∞)=1 . (3) F(x,y)关于每个变量都是右连续的,即 F(x+0,y)= F(x,y), F(x,y+0)= F(x,y) . (4)对于任意实数x 1 P{x 1 二.二维离散型随机变量及其联合分布律 1.定义 若随机变量(X,Y)只能取有限对或可列无限多对值(x i ,y j ) (i ,j =1,2,… )称(X,Y)为二维离散型随机变量.并称P{X= x i ,Y= y j }= p i j 为(X,Y)的联合分布律.也可列表表示. 2.性质 (1)非负性 0≤p i j ≤1 . (2)归一性 ∑∑=i j ij p 1 . 3. (X,Y)的(X 和Y 的联合)分布函数F(x,y)=∑ ∑≤≤x x y y ij i j p 三.二维连续型随机变量及其联合概率密度 1.定义 如果存在非负的函数f (x,y),使对任意的x 和y,有F(x,y)=??∞-∞-y x dudv v u f ),( 则称(X,Y)为二维连续型随机变量,称f(x,y)为(X,Y)的(X 和Y 的联合)概率密度. 2.性质 (1)非负性 f (x,y)≥0 . (2)归一性 1),(=??∞ ∞-∞ ∞-d x d y y x f . (3)若f (x,y)在点(x,y)连续,则y x y x F y x f ???=),(),(2 (4)若G 为xoy 平面上一个区域,则??=∈G dxdy y x f G y x P ),(}),{(. 四.边缘分布 1. (X,Y)关于X 的边缘分布函数 F X (x) = P{X ≤x , Y<∞}= F (x , ∞) . (X,Y)关于Y 的边缘分布函数 F Y (y) = P{X<∞, Y ≤y}= F (∞,y) 2.二维离散型随机变量(X,Y) 关于X 的边缘分布律 P{X= x i }= ∑∞ =1 j ij p = p i · ( i =1,2,…) 归一性 11 =∑∞ =?i i p . 关于Y 的边缘分布律 P{Y= y j }= ∑∞=1 i ij p = p ·j ( j =1,2,…) 归一性 11 =∑∞ =?j j p . 3.二维连续型随机变量(X,Y) 关于X 的边缘概率密度f X (x)=?∞∞-dy y x f ),( 归一性1)(=?∞ ∞-dx x f X 关于Y 的边缘概率密度f Y (y)=x d y x f ?∞ ∞-),( 归一性1)(=?∞ ∞-dy y f Y 五.相互独立的随机变量 1.定义 若对一切实数x,y,均有F(x,y)= F X (x) F Y (y) ,则称X 和Y 相互独立. 2.离散型随机变量X 和Y 相互独立?p i j = p i ··p ·j ( i ,j =1,2,…)对一切x i ,y j 成立. 3.连续型随机变量X 和Y 相互独立?f (x,y)=f X (x)f Y (y)对(X,Y)所有可能取值(x,y)都成立. 六．条件分布 1．二维离散型随机变量的条件分布 定义 设(X,Y)是二维离散型随机变量,对于固定的j,若P{Y=y j }>0,则称 },{j i j i p y Y x X P == P{X=x i |Y=y j } 为在Y= y j 条件下随机变量X 的条件分布律. 同样,对于固定的i,若P{X=x i }>0,则称 P{Y=y j |X=x i } 为在X=x i 条件下随机变量Y 的条件分布律. 第四章 随机变量的数字特征 一.数学期望和方差的定义 随机变量X 离散型随机变量 连续型随机变量 分布律P{X=x i }= p i ( i =1,2,…) 概率密度f (x) 数学期望(均值)E(X) ∑∞ =1 i i i p x (级数绝对收敛) ?∞ ∞-dx x xf )((积分绝对收敛) 方差D(X)=E{[X-E(X)]2 } []∑-∞ =1 2 )(i i i p X E x ?-∞ ∞-dx x f X E x )()]([2 =E(X 2)-[E(X)]2 (级数绝对收敛) (积分绝对收敛) 函数数学期望E(Y)=E[g(X)] i i i p x g ∑∞ =1)((级数绝对收敛) ?∞ ∞-dx x f x g )()((积分绝对收敛) 标准差σ(X)=√D(X) . 二.数学期望与方差的性质 1. c 为为任意常数时, E(c) = c , E(cX) = cE(X) , D(c) = 0 , D (cX) = c 2 D(X) . 2.X,Y 为任意随机变量时, E (X ±Y)=E(X)±E(Y) . 3. X 与Y 相互独立时, E(XY)=E(X)E(Y) , D(X ±Y)=D(X)+D(Y) . 4. D(X) = 0 ? P{X = C}=1 ,C 为常数. 三.六种重要分布的数学期望和方差 E(X) D(X) 1.X~ (0-1)分布P{X=1}= p (0 ,}{},{? =====i j i i j i p p x X P y Y x X P 2.X~ b (n,p) (0 3.X~ π(λ) λ λ 4.X~ U(a,b) (a+b)/2 (b-a) 2/12 5.X 服从参数为θ的指数分布 θ θ2 6.X~ N (μ,σ2) μ σ2 四.矩的概念 随机变量X 的k 阶(原点)矩E(X k ) k=1,2,… 随机变量X 的k 阶中心矩E{[X-E(X)] k } 随机变量X 和Y 的k+l 阶混合矩E(X k Y l ) l=1,2,… 随机变量X 和Y 的k+l 阶混合中心矩E{[X-E(X)] k [Y-E(Y)] l } 第六章 样本和抽样分布 一.基本概念 总体X 即随机变量X ; 样本X 1 ,X 2 ,…,X n 是与总体同分布且相互独立的随机变量;样本值x 1 ,x 2 ,…,x n 为实数;n 是样本容量. 统计量是指样本的不含任何未知参数的连续函数.如： 样本均值∑==n i i X n X 1 1 样本方差() ∑--==n i i X X n S 12211 样本标准差S 样本k 阶矩∑==n i k i k X n A 1 1( k=1,2,…) 样本k 阶中心矩∑-==n i k i k X X n B 1)(1( k=1,2,…) 二.抽样分布 即统计量的分布 1.X 的分布 不论总体X 服从什么分布, E (X ) = E(X) , D (X ) = D(X) / n . 特别,若X~ N (μ,σ2 ) ,则 X ~ N (μ, σ2 /n) . 2.χ2 分布 (1)定义 若X ～N (0,1 ) ,则Y =∑=n i i X 1 2~ χ2(n)自由度为n 的χ2分布. (2)性质 ①若Y~ χ2(n),则E(Y) = n , D(Y) = 2n . ②若Y 1~ χ2(n 1) Y 2~ χ2(n 2) ,则Y 1+Y 2~ χ2(n 1 + n 2). ③若X~ N (μ,σ2 ), 则 2 2 )1(σS n -~ χ2(n-1),且X 与S 2相互独立. (3)分位点 若Y~ χ2(n),0< α <1 ,则满足 αχχχχαααα=<>=<=>--))}(())({()}({)}({2 2/122/212n Y n Y P n Y P n Y P 的点)()(),(),(22 /122/212n n n n ααααχχχχ--和分别称为χ2 分布的上、下、双侧α分位点. 3. t 分布 (1)定义 若X~N (0,1 ),Y~ χ2 (n),且X,Y 相互独立,则t=n Y X ~t(n)自由度为n 的t 分布. (2)性质①n →∞时,t 分布的极限为标准正态分布. ②X ～N (μ,σ2 )时, n S X μ -~ t (n-1) . ③两个正态总体 相互独立的样本 样本均值 样本方差 X~ N (μ1,σ12 ) 且σ12=σ22=σ2 X 1 ,X 2 ,…,X n1 X S 12 Y~ N (μ2,σ22 ) Y 1 ,Y 2 ,…,Y n2 Y S 22 则 2 12111)()(n n S Y X w + ---μμ~ t (n 1+n 2-2) , 其中 2)1()1(212 2 2211 2-+-+-=n n S n S n S w (3)分位点 若t ~ t (n) ,0 < α<1 , 则满足 αααα=>=-<=>)}({)}({)}({2/n t t P n t t P n t t P 的点)(),(),(2/n t n t n t ααα±-分别称t 分布的上、下、双侧α分位点. 注意: t 1- α (n) = - t α (n). 4.F 分布 (1)定义 若U~χ2 (n 1), V~ χ2 (n 2), 且U,V 相互独立,则F =2 1 n V n U ~F(n 1,n 2)自由度为 (n 1,n 2)的F 分布. (2)性质(条件同3.(2)③) 22 21 2221σσS S ~F(n 1-1,n 2-1) (3)分位点 若F~ F(n 1,n 2) ,0< α <1,则满足 )},({)},({21121n n F F P n n F F P αα-<=> ααα=<>=-))},(()),({(212/1212/n n F F n n F F P 的点),(),(),,(),,(212/1212/21121n n F n n F n n F n n F αααα--和分别称为F 分布的上、下、双侧α 分位点. 注意: .) .(1 ),(12211n n F n n F αα= - 第七章 参数估计 一.点估计 总体X 的分布中有k 个待估参数θ1, θ2,…, θk . X 1 ,X 2 ,…,X n 是X 的一个样本, x 1 ,x 2 ,…,x n 是样本值. 1.矩估计法 先求总体矩?????===),,,(),,,(),,,(2121222111k k k k k θθθμμθθθμμθθθμμ 解此方程组,得到??? ??===),,,(),,,(),,,(2121222111k k k k k μμμθθμμμθθμμμθθ , 以样本矩A l 取代总体矩μ l ( l=1,2,…,k)得到矩估计量??? ????===∧ ∧ ∧ ),,,(),,,(),,,(2121222111k k k k k A A A A A A A A A θθθθθθ, 若代入样本值则得到矩估计值. 2.最大似然估计法 若总体分布形式(可以是分布律或概率密度)为p(x, θ1, θ2,…, θk ),称样本X 1 ,X 2 ,…,X n 的联合分布∏==n i k i k x p L 12121),,,,(),,,(θθθθθθ 为似然函数.取使似然函数达到最大值的 ∧ ∧∧k θθθ,,,21 ,称为参数θ1, θ2,…,θk 的最大似然估计值,代入样本得到最大似然估计量. 若L(θ1, θ2,…, θk )关于θ1, θ2,…, θk 可微,则一般可由 似然方程组 0=??i L θ 或 对数似然方程组 0ln =??i L θ (i =1,2,…,k) 求出最大似然估计. 3.估计量的标准 (1) 无偏性 若E(∧θ)=θ,则估计量∧ θ称为参数θ的无偏估计量. 不论总体X 服从什么分布, E (X )= E(X) , E(S 2)=D(X), E(A k )=μk =E(X k ),即样本均值X , 样本方差S 2,样本k 阶矩A k 分别是总体均值E(X),方差D(X),总体k 阶矩μk 的无偏估计, (2)有效性 若E(∧θ1 )=E(∧θ2)= θ, 而D(∧θ1)< D(∧θ2), 则称估计量∧θ1比∧ θ2有效. (3)一致性(相合性) 若n →∞时,θθP →∧,则称估计量∧ θ是参数θ的相合估计量. 二.区间估计 1.求参数θ的置信水平为1-α的双侧置信区间的步骤 (1)寻找样本函数W=W(X 1 ,X 2 ,…,X n ,θ),其中只有一个待估参数θ未知,且其分布完全确定. (2)利用双侧α分位点找出W 的区间(a,b),使P{a 2.单个正态总体 待估参数 其它参数 W 及其分布 置信区间 μ σ2 已知 n X σμ -~N (0,1) (2/ασz n X ±) μ σ2未知 n S X μ -~ t (n-1) )1((2/-±n t n S X α σ2 μ未知 2 2 )1(σS n -~ χ2(n-1) )) 1()1(,)1()1((22/12 2 2/2 -----n S n n S n ααχχ 3.两个正态总体 (1)均值差μ 1-μ 2 其它参数 W 及其分布 置信区间 已知 22 21,σσ 2 22 1 2 1 21) (n n Y X σσμμ+ --- ~ N(0,1) )(2 22 1 21 2 n n z Y X σσα + ±- 未知2 2221σσσ== 2 12111)(n n S Y X w +---μμ～t(n 1+n 2-2) )1 1)2((21212n n S n n t Y X w +-+±-α 其中S w 等符号的意义见第六章二. 3 (2)③. (2) μ 1,μ 2未知, W= 22 212221σσS S ~ F(n 1-1,n 2-1),方差比σ12/σ22的置信区间为 )) 1,1(1,)1,1(1(212/1222 1212/2221----?-n n F S S n n F S S αα 注意:对于单侧置信区间,只需将以上所列的双侧置信区间中的上(下)限中的下标α/2改为α,另外的下(上)限取为-∞ (∞)即可. 第一章 随机事件及其概率 知识点：概率的性质 事件运算 古典概率 事件的独立性 条件概率 全概率与贝叶斯公式 常用公式 )()()()()()2(加法定理AB P B P A P B A P -+=Y ) ,,() ()(2111有限可加性两两互斥设n n i i n i i A A A A P A P ΛY ∑===) ,(0)()() ()()(互不相容时独立时与B A AB P B A B P A P AB P ==)()()()()5(AB P A P B A P B A P -==-)() ()()()(时当A B B P A P B A P B A P ?-==-))0(,,() ()/()()()6(211 >Ω=∑=i n n i i i A P A A A A B P A P B P 且的一个划分为其中全概率公式Λ) ,,()] (1[1)(2111相互独立时n n i i n i i A A A A P A P ΛY ∏==--=) /()()/()()()4(B A P B P A B P A P AB P ==) (/)()/()3(A P AB P A B P =) ()/()()/()()/()7(1逆概率公式∑==n i i i i i i A B P A P A B P A P B A P )(/)()(/)()1(S L A L A P n r A P == 应用举例 1、已知事件,A B 满足)()(B A P AB P =，且6.0)(=A P ，则=)(B P （ ）。 2、已知事件,A B 相互独立，,)(k A P =6.0)(,2.0)(==B A P B P Y ，则= k （ ）。 3、已知事件,A B 互不相容，,3.0)(=A P ==)(,5.0)(B A P B P Y 则（ ）。 4、若,3.0)(=A P ===)(,5.0)(,4.0)(B A B P B A P B P Y （ ）。 5、,,A B C 是三个随机事件，C B ?，事件()A C B -U 与A 的关系 是（ ）。 6、5张数字卡片上分别写着1，2，3，4，5，从中任取3 张，排成3位数，则排成3位奇数的概率是（ ）。 7、某人下午5:00下班。他所积累的资料表明： 某日他抛一枚硬币决定乘地铁还是乘汽车。 （1）试求他在5:40~5:50到家的概率； （2）结果他是5:47到家的。试求他是乘地铁回家的概率。 解（1）设1A ={他是乘地铁回家的}，2A ={他是乘汽车回家的}， i B ={第i 段时间到家的}，4,3,2,1=i 分别对应时间段 5:30~5:40，5:40~5:50，5:50~6:00，6:00以后 则由全概率公式有 )|()()|()()(2221212A B P A P A B P A P B P += 由上表可知4.0)|(12=A B P ，3.0)|(22=A B P ，5.0)()(21==A P A P 浙江省2018年1月高等教育自学考试 英国文学选读试题 课程代码：10054 Part I. Choose the relevant match from Column B for each item in Column A. (10%) Section A A B (1)Jonathan Swift（） A. The Rainbow (2)D.H. Lawrence （） B. Adam Bede (3)Emily Brontё（） C. Gulliver’s Travels (4)Thomas Hardy （） D. Wuthering Heights (5)George Eliot（） E. Far From the Madding Crowd Section B A B (1) Middlemarch（） A. Shylock (2) Jane Eyre（） B. Sir Peter Teazle (3) The Merchant of Venice（） C. Mr. Rochester (4) Mrs. Warren’s Profession（） D. Will Ladislas (5) The School for Scandal（） E. Vivie Part II. Complete each of the following statements with a proper word or a phrase according to the textbook. (5%) 1. In Paradise Lost, the author intended to expose the ways of Satan and to “justify the ways of _________ to men.” 2. As the greatest novelist of the Victorian period, Charles Dickens set out a full map, and a large -scale criticism of the _________century. 3. In Jane Austen’s novels, stories of _________ and marriage provide the major themes. 4. In the novel Tess of the D’Urbervilles, the two men Alec and _________ are both agents of the destructive force of the society. 1 第1章盎格鲁-撒克逊时期(450～1066) 1.1 复习笔记 Ⅰ. Historical Background（历史背景） (1) The earliest settlers of the British Isles were the Celts, who migrated to the British Isles about 600 B.C. 不列颠群岛最早的定居者是凯尔特人，他们大约在公元前600年移民到不列颠群岛。 (2) From 55 B.C. to 407 A.D. the British Isles were under the rule of the Roman Empire. 从公元前55年到公元407年，不列颠群岛处于罗马帝国的统治之下。 (3) About 450 A.D., waves of the Angles, Saxons, and Jutes invaded the British Isles. They settled in England, and drove the Celts into Wales, Scotland, and Ireland. 公元450年左右，盎格鲁人、撒克逊人和朱特人占领了不列颠群岛。他们在英格兰定居，将凯尔特人驱逐到威尔士、苏格兰和爱尔兰。 (4) It was around 500 A.D., in the struggle against Cerdic, the founder of the kingdom of Wessex, that the Celtic King Arthur, a legendary figure, is said to have acquired his fame. 大约在公元500年，在与威塞克斯王国创始人塞迪奇的斗争中，传说中的凯尔特王亚瑟获得了他的名声。 (5) Beginning from the later part of the 8th century, the Danes, or the Vikings, came to invade England, at first, along the eastern coast, but later they threatened 第一章 随机事件与概率 第一节 随机事件及其运算 1、 随机现象：在一定条件下，并不总是出现相同结果的现象 2、 样本空间：随机现象的一切可能基本结果组成的集合，记为Ω={ω},其中ω 表示基本结果，又称为样本点。 3、 随机事件：随机现象的某些样本点组成的集合常用大写字母A 、B 、C 等表 示，Ω表示必然事件， ?表示不可能事件。 4、 随机变量：用来表示随机现象结果的变量，常用大写字母X 、Y 、Z 等表示。 5、 时间的表示有多种： （1） 用集合表示，这是最基本形式 （2） 用准确的语言表示 （3） 用等号或不等号把随机变量于某些实属联结起来表示 6、事件的关系 （1）包含关系：如果属于A 的样本点必属于事件B ，即事件 A 发生必然导致事 件B 发生，则称A 被包含于B ，记为A ?B; （2）相等关系：若A ?B 且B ? A ，则称事件A 与事件B 相等，记为A ＝B 。 （3）互不相容：如果A ∩B= ?，即A 与B 不能同时发生，则称A 与B 互不相容 7、事件运算 （1）事件A 与B 的并：事件A 与事件B 至少有一个发生，记为 A ∪B 。 （2）事件A 与B 的交：事件A 与事件B 同时发生，记为A∩ B 或AB 。 （3）事件A 对B 的差：事件A 发生而事件B 不发生,记为 A －B 。用交并补可以 表示为B A B A =-。 （4）对立事件：事件A 的对立事件（逆事件），即 “A 不发生”，记为A 。 对立事件的性质：Ω=?Φ=?B A B A ,。 8、事件运算性质：设A ，B ，C 为事件，则有 （1）交换律：A ∪B=B ∪A ，AB=BA （2）结合律：A ∪(B ∪C)=(A ∪B)∪C=A ∪B ∪C A(BC)=(AB)C=ABC （3）分配律：A ∪(B∩C)＝(A ∪B)∩(A∪C)、 A(B ∪C)＝(A∩B)∪(A∩C)= AB ∪AC （4）棣莫弗公式（对偶法则）：B A B A ?=? B A B A ?=? 9、事件域：含有必然事件Ω ，并关于对立运算和可列并运算都封闭的事件类ξ 称为事件域，又称为σ代数。具体说，事件域ξ满足： （1）Ω∈ξ; (2)若A ∈ξ，则对立事件A ∈ξ； （3）若A n ∈ξ，n=1,2，···，则可列并 ∞ =1 n n A ∈ξ 。 Hamlet is the first work of literature to look squarely at the stupidity, falsity and sham of everyday life, without laughing and without easy answers. In a world where things are not as they seem, Hamlet…s genuineness, thoughtfulness, and sincerity make him special. Hamlet is no saint. But unlike most of the other characters (and most people today), Hamlet chooses not to compromise with evil. Dying, Hamlet reaffirms the tragic dignity of a basically decent person in a bad world Hamlet is the first work of literature to show an ordinary person looking at the futility and wrongs in life, asking the toughest questions and coming up with honest semi-answers like most people do today. Unlike so much of popular culture today, "Hamlet" leaves us with the message that life is indeed worth living, even by imperfect people in an imperfect world. 犹豫scholars have debated for centuries about Hamlet's hesitation in killing his uncle. Some see it as a plot device to prolong the action, and others see it as the result of pressure exerted by the complex philosophical and ethical issues that surround cold-blooded murder, calculated revenge and thwarted desire. More recently, psychoanalytic critics have examined Hamlet's unconscious desires (Freud concludes that Hamlet has an "Oedipal desire for his mother and the subsequent guilt [is] preventing him from murdering the man [Claudius] who has done what he unconsciously wanted to do". Robinson Crusoe is a grand hero in westerners? eyes. He survived in the deserted island and lived a meaningful life. He almost has everything needed for becoming a successful man, such as his excellent creativity, great working capacity, courage, and persistence in overcoming obstacles. But he has shortcomings, too. Sometimes he was irresolute; he was not confident; he was fetishistic, although his belief had done him much good. He serves somehow as a lighthouse for the ambitious people. It?s also instructive for average people. Robinson was the representative of the bourgeois of the 18th C. It was the time when bourgeois grew stronger and stronger. The author Defoe paid a tribute to bourgeois by creating such a rational, powerful, clever, and successful man. 【Themes of Robinson Crusoe】 1. The ambivalence of mastery In short, while Crusoe seems praiseworthy in mastering his fate by overcoming his obstacles, and controlling his environment, the praiseworthiness of his mastery over his fellow human Friday is more doubtful. Defoe explores the link between the two in his depiction of the colonial mind. 2. The necessity of Repentance Crusoe?s experiences constitute not simply an adventure story in which thrilling things happen, but also a moral tale illustrating the right and wrong ways to live one?s life. Crusoe?s story instruct s others in God?s wisdom, and one vital part of this wisdom is the importance of repenting one?s sins. 3. The Importance of Self-Awareness Crusoe?s arrival on the island does not make him revert to a brute existence and he remains conscious of himself at all times. His island existence actually deepens his self-awareness as he withdraws from the external society and turns inward. The idea that the individual must keep a careful reckoning of the state of his own soul is a key point in the Presbyterian doctrine that the aothor took seriously all his life. ·Jane Eyre Charlotte Bronte The protagonist and title character, orphaned as a baby. She is a plain-featured, small and reserved but talented, sympathetic, hard-working, honest and passionate girl. Skilled at studying, drawing, and teaching, she works as a governess at Thornfield Hall and falls in love with her wealthy employer, Edward Rochester. But her strong sense of conscience does not permit her to become his mistress, and she does not return to him until his insane wife is dead and she herself has come into an inheritance. 【Themes of Jane Eyre】 1. Gender relations A particularly important theme in the novel is patriarchalism and Jane…s efforts to assert her own identity within male-dominated society. Among the three of the main male characters, Geoffrey Chaucer (1343-1400) 乔叟He was born in 1343 in London. He died in 1400 and was buried in Westminster Abbey, thus founding the “Poets Corner”.The father of English Poetry and one of the greatest narrative poets of England.“The Canterbury Tales” (1387-1400) It is Chaucer?s masterpiece and one of the monumental works in English literature. Chaucer’s Contribution to English Literature Chaucer is regarded as the founder of English poetry and has been called “the founder of English realism.” He is the firs t great poet who wrote in the English language. He introduced from France the rhymed stanza of various types, especially the “heroic couplet” (英雄双韵体) to English poetry.His masterpiece “The Canterbury T ales” is one of the monumental works in English literature 公爵夫人之书，百鸟议会，声誉之堂，特罗勒思和克里西德 Structure of a poem: A poem can be broken down into three parts: (1) Stanza (节) : a group of lines set off from the other lines in a poem. It is the poetic equivalent of a paragraph in prose. In traditional poems, the stanza usually contains a unit of thought.(2) The line (行) : a single line of poetry (3) The foot (音步) : a syllable or a group of 2 or 3 syllables. T o scan a line of poetry one counts the number of feet in a line. For a beginner, the easiest thing to do is to count the number of stresses. Typically a foot will contain a stressed and an unstressed syllable. William Shakespeare (1564-1616)playwright, poet, actor.Shakespeare and Aeschylus are the two greatest dramatic geniuses the world has ever known.—Carl Marks.The Great Tragedies: 《哈姆雷特》(Hamlet,1601 ) 《奥赛罗》(Othello, 1604) 《李尔王》(King Lear, 1605) 《麦克白》(Macbeth, 1606) The Great Comedies威尼斯商人》(The Merchant of Venice, 1596) 《仲夏夜之梦》(A Midsummer Night's Dream,1596) 《第十二夜》(Twelfth Night, 1600) 《皆大欢喜》(As You Like It, 1601) Shakespeare’s car eer as a dramatist may be divided into four major phases.: The First Period(1590-1594) This period is the period of his apprenticeship in play-writing. Works: Henry VI The Comedy of Errors《错误的喜剧》/《连环错》Love?s Labor?s Lost 《迷失的爱》/《空爱一场》/《爱的徒劳》Romeo and Juliet, etc. The Second Period (1595-1600) This period is his mature period, mainly a period of “great comedies” and mature historical plays. It includes 6 comedies, 5 historical plays and 1 Roman tragedy. His sonnets are also thought to be written in this period. The Third Period (1601-1607) The third period of Shakespeare?s dramatic career is mainly the period of “great tragedies” and “dark comedies”. It includes 5 tragedies, 3 comedies and 2 Roman tragedies.Major works written in this period:Hamlet, Othello, King Lear, Macbeth, Antony and Cleopatra The Fourth Period (1608-1612) The fourth period of Shakespeare?s work is the period of romantic drama. It includes 4 romances or “reconciliation（和解，复合）plays”. Shakespeare’s Literary Position:Shakespeare and the Authorized Version of the English Bible are the two greatest treasuries of the English language. Shakespeare has been universally acknowledged to be the summit of the English Renaissance, and one of the greatest writers in world literature. Hamlet:Hamle t is considered the summit of Shakespeare?s art. It is one of Shakespeare?s canon, and it is universally included in the list of the world?s greatest works.It?s written in the form of blank verse.blank verse : poetry in rhymeless iambic pentameter.（素体诗剧）The story, coming from an old Danish legend, is a tragedy of the “revenge” genre. Shakespeare incorporates into the medieval story other major humanistic themes, including love, justice, good and evil, and most notably, madness, and the spirit of the time Injustice, conspiracy, and betrayal in the society。1. first blow: father?s murder and mother?s re-marriage2.second blow: betrayal of his two former friends Rosencrantz and Guildenstern3. third blow: betrayal of his girl friend OpheliaThe greatness o f the play: in praise of the noble quality of Prince Hamlet as a representative of humanist thinkers and his disillusionment with the corrupt and degenerated society in which he lived. 做试题,没答案?上自考365,网校名师为你详细解答! 浙江省2007年10月高等教育自学考试 英国文学选读试题 课程代码：10054 Ⅰ.Choose the relevant match from Column B for each item in Column A.(10%) Section A A B (1)Daniel Defoe ( ) A. The Pilgrim’s Progress (2)Charles Dickens ( ) B. The Silver Box (3)John Bunyan ( ) C. Robinson Crusoe (4)Richard Sheridan ( ) D. A Tale of Two Cities (5)John Galsworthy ( ) E. The School for Scandal Section B A B (1) Jane Eyre( ) A. Irene (2) The Man of Property( ) B. Mr. Rochester (3) The Merchant of Venice( ) C. Satan (4) Paradise Lost( ) D. Sophia Western (5) The History of Tom Jones ( ) E. Portia Ⅱ.Complete each of the following statements with a proper word or a phrase according to the textbook.(5%) 1. In the era of the Renaissance, the humanists made attempts to get rid of those old ______ ideas in medieval Europe. 2. The ______ century was an age of prose. A group of excellent writers, such as Swift, Fielding were produced. 3. English ______ is generally said to have begun in 1798 with the publication of Wordsworth and Coleridge’s Lyrical Ballads. 4. In the V ictorian period, the______ as a literary genre became the most widely read and the most vital and challenging expression of progressive thought. 5. The Waste Land by T.S. Eliot is a poem concerned with the spiritual breakup of a modern 1 Chapter 8 The age of Romanticism: 1.From the publication of Lyrical Ballads by Wordsworth and Coleridge in 1798 to the death of Sir Walter Scott in 1832, a new movement appeared on the literary arena. The essence of this new movement is the glorification of instinct and emotion, a deep veneration of nature, and a flaming zeal to remark the world. 2.The political and social factors that gave rise to the romantic movement were the three revolution: American and French Revolution; national liberation movements; democratic movements. 3.And Industrial Revolution: brought great wealth to the rich and worsened the living condition of the poor; Workers organized themselves and gave voice to their distress by breaking machines, which is called Machine breaking movement(Luddite movement) 4.The shift in literature from emphasis on reason to instinct and emotion was intellectually prepared for by a number of thinkers in the later half of the 18th century. 5.Rousseau: the father of Romanticism. He rejects the worship of reason. He maintains in the really vital problems of life, it is much safer to rely on feelings, to follow our instincts and emotions. He preaches that civilized man should return to nature, praised the natural man as the noble savage and attacks the civilized man as the depraved animals. 6.Edmund Burke: As a political philosopher he is known for his Reflection on the Revolution in France. He distinguished between two kinds of beauty—the sublime and the beautiful. 7.Thomas Paine: He published The Rights of Man in 1791 to answer to Burke’s Reflection. The Rights of Man asserts that man has no property in man and justifies the radical actions of French people in the revolution, claiming that it is the right of people to overthrown a government that opposes humanity. 8.Characteristic features of the romantic movement: 1)Subjectivism: romantic poets describe poetry as the spontaneous overflow of powerful feelings, which express the poet’s mind.The interest of romantic poets is not objective world or the action of men, but in the feelings, thoughts, and experience of the poets themselves. 2)Spontaneity: Wordsworth defines poetry as the spontaneous overflow of feelings. Romanticism is an assertion of independence, a departure from the neo-classis rules. 3)Singularity 4)Worship of nature 5)Simplicity 6)Melancholy 7)It was an age of poetry by which the poets outpoured their feelings and emotions. Romantic poets loved to use a freer verse form. 9.Romanticism is a term that denotes most of the writings that were written between 1798 and 1832. Wordsworth: 概率论与数理统计主要内容小结 概率部分 1、全概率公式与贝叶斯公式 全概率公式： )()|()(11B P B A P A P = ++)()|(22B P B A P )()|(n n B P B A P + 其中n B B B ,,,21 是空间S 的一个划分。 贝叶斯公式：∑== n j j j i i i B A P B P B A P B P A B P 1 ) |()() |()()|( 其中n B B B ,,,21 是空间S 的一个划分。 2、互不相容与互不相关 B A ,互不相容0)(,==?B A P B A φ 事件B A ,互相独立))(()(B A P B A P =? ; 两者没有必然联系 3、几种常见随机变量概率密度与分布律:两点分布，二项分布，泊松分布，均匀分布，二项分布，指数分布，正态分布。 ),,1(~p b X 即二点分布，则分布律为.1,0,)1(}{1=-==-k p p k x P k k ),,(~p n b X 即二项分布，则分布律为.,...,1,0,)1(}{n k p p C k x P k n k k n =-==- ),(~λπX 即泊松分布，则分布律为,......1,0,! }{== =-k k e k x P k λ λ ),,(~b a U X 即均匀分布，则概率密度为.,0),(,1 )(??? ??∈-=其它 b a x a b x f ),(~θE X 即指数分布，则概率密度为.,00 ,1)(?? ???>=-其它x e x f x θ θ ),,(~2σμN X 即正态分布，则则概率密度为+∞<<-∞= - x e x f x ,21)(2 2π . 概率论和数理统计真题讲解 （一）单项选择题（本大题共10小题，每小题2分，共20分） 在每小题列出的四个备选项中只有一个是符合题目要求的，请将其代码填写在题后的括号内。错选、多选或未选均无分。 1.设随机事件A与B互不相容，且P（A）＞0,P（B）＞0，则（） A.P（B|A）＝0 B.P（A|B）＞0 C.P（A|B）＝P（A） D.P（AB）＝P（A）P（B） 『正确答案』分析：本题考察事件互不相容、相互独立及条件概率。 解析：A：，因为A与B互不相容，，P（AB）＝0，正确； 显然，B，C不正确；D：A与B相互独立。 故选择A。 提示：① 注意区别两个概念：事件互不相容与事件相互独立； ② 条件概率的计算公式：P（A）＞0时，。 2.设随机变量X～N（1，4），F（x）为X的分布函数，Φ（x）为标准正态分布函数，则F（3）＝（） A.Φ（0.5） B.Φ（0.75） C.Φ（1） D.Φ（3） 『正确答案』分析：本题考察正态分布的标准化。 解析：， 故选择C。 提示：正态分布的标准化是非常重要的方法，必须熟练掌握。 3.设随机变量X的概率密度为f（x）＝则P{0≤X≤}＝（） 『正确答案』分析：本题考察由一维随机变量概率密度求事件概率的方法。第33页 解析：， 故选择A。 提示：概率题目经常用到“积分的区间可加性”计算积分的方法。 4.设随机变量X的概率密度为f（x）＝则常数c＝（） A.－3 B.－1 C.－ D.1 『正确答案』分析：本题考察概率密度的性质。 解析：1＝，所以c＝－1， 故选择B。 提示：概率密度的性质： 1.f（x）≥0； 4.在f（x）的连续点x，有F′（X）＝f（x）；F（x）是分布函数。课本第38页 5.设下列函数的定义域均为（－∞，＋∞），则其中可作为概率密度的是（） A.f（x）＝－e－x B. f（x）＝e－x C. f（x）＝ D.f（x）＝ 『正确答案』分析：本题考察概率密度的判定方法。 解析：① 非负性：A不正确；② 验证：B：发散； C：，正确；D：显然不正确。 故选择C。 提示：判定方法：若f（x）≥0，且满足，则f（x）是某个随机变量的概率密度。 6.设二维随机变量（X，Y）～N（μ1,μ2，），则Y ～（） 『正确答案』分析：本题考察二维正态分布的表示方法。 解析：显然，选择D。大学概率论与数理统计的复习资料
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