机器学习的数学基础（下）

矩阵的特征值和特征向量

**1.矩阵的特征值和特征向量的概念及性质 **

(1) 设 λ \lambda λ是 A A A的一个特征值，则
kA , aA + bE , A 2 , A m , f ( A ) , A T , A − 1 , A ∗ \text{kA},\text{aA} + \text{bE},A^{2},A^{m},f(A),A^{T},A^{- 1},A^{\ast} kA,aA+bE,A2,Am,f(A),AT,A−1,A∗有一个特征值分别为
kλ , aλ + b , λ 2 , λ m , f ( λ ) , λ , λ − 1 , ∣ A ∣ λ , \text{kλ},\text{aλ} + b,\lambda^{2},\lambda^{m},f(\lambda),\lambda,\lambda^{- 1},\frac{|A|}{\lambda}, kλ,aλ+b,λ2,λm,f(λ),λ,λ−1,λ∣A∣,且对应特征向量相同（ A T A^{T} AT
例外）。

(2)
若 λ 1 , λ 2 , ⋯ , λ n \lambda_{1},\lambda_{2},\cdots,\lambda_{n} λ1,λ2,⋯,λn为 A A A的 n n n个特征值，则 ∑ i = 1 n λ i = ∑ i = 1 n a ii , ∏ i = 1 n λ i = ∣ A ∣ \sum_{i = 1}^{n}\lambda_{i} = \sum_{i = 1}^{n}a_{\text{ii}},\prod_{i = 1}^{n}\lambda_{i} = |A| ∑i=1nλi=∑i=1naii,∏i=1nλi=∣A∣
,从而 ∣ A ∣ ≠ 0 ⇔ A |A| \neq 0 \Leftrightarrow A ∣A∣=0⇔A没有特征值。

(3)
设 λ 1 , λ 2 , ⋯ , λ s \lambda_{1},\lambda_{2},\cdots,\lambda_{s} λ1,λ2,⋯,λs为 A A A的 s s s个特征值，对应特征向量为
α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs，

若:
α = k 1 α 1 + k 2 α 2 + ⋯ + k s α s \alpha = k_{1}\alpha_{1} + k_{2}\alpha_{2} + \cdots + k_{s}\alpha_{s} α=k1α1+k2α2+⋯+ksαs
,

则:
A n α = k 1 A n α 1 + k 2 A n α 2 + ⋯ + k s A n α s = k 1 λ 1 n α 1 + k 2 λ 2 n α 2 + ⋯ k s λ s n α s A^{n}\alpha = k_{1}A^{n}\alpha_{1} + k_{2}A^{n}\alpha_{2} + \cdots + k_{s}A^{n}\alpha_{s} = k_{1}\lambda_{1}^{n}\alpha_{1} + k_{2}\lambda_{2}^{n}\alpha_{2} + \cdots k_{s}\lambda_{s}^{n}\alpha_{s} Anα=k1Anα1+k2Anα2+⋯+ksAnαs=k1λ1nα1+k2λ2nα2+⋯ksλsnαs
。

**2.相似变换、相似矩阵的概念及性质 **

(1) 若 A ∼ B A \sim B A∼B，则

1) A T ∼ B T , A − 1 ∼ B − 1 , , A ∗ ∼ B ∗ A^{T} \sim B^{T},A^{- 1} \sim B^{- 1},,A^{\ast} \sim B^{\ast} AT∼BT,A−1∼B−1,,A∗∼B∗

∣ A ∣ = ∣ B ∣ , ∑ i = 1 n A ii = ∑ i = 1 n b ii , r ( A ) = r ( B ) |A| = |B|,\sum_{i = 1}^{n}A_{\text{ii}} = \sum_{i = 1}^{n}b_{\text{ii}},r(A) = r(B) ∣A∣=∣B∣,∑i=1nAii=∑i=1nbii,r(A)=r(B)

3) ∣ λ E − A ∣ = ∣ λ E − B ∣ |\lambda E - A| = |\lambda E - B| ∣λE−A∣=∣λE−B∣，对 ∀ λ \forall\lambda ∀λ成立

**3.矩阵可相似对角化的充分必要条件 **

(1)
设 A A A为 n n n阶方阵，则 A A A可对角化 ⇔ \Leftrightarrow ⇔对每个 k i k_{i} ki重根特征值 λ i \lambda_{i} λi，有 n − r ( λ i E − A ) = k i n - r(\lambda_{i}E - A) = k_{i} n−r(λiE−A)=ki

(2)
设 A A A可对角化，则由 P − 1 AP = Λ , P^{- 1}\text{AP} = \Lambda, P−1AP=Λ,有 A = PΛ P − 1 A = \text{PΛ}P^{- 1} A=PΛP−1，从而 A n = P Λ n P − 1 A^{n} = P\Lambda^{n}P^{- 1} An=PΛnP−1

(3) 重要结论

1) 若 A ∼ B , C ∼ D A \sim B,C \sim D A∼B,C∼D，则 [ A O O C ] ∼ [ B O O D ] \begin{bmatrix} & A\quad O \\ & O\quad C \\ \end{bmatrix} \sim \begin{bmatrix} & B\quad O \\ & O\quad D \\ \end{bmatrix} [AOOC]∼[BOOD].

若 A ∼ B A \sim B A∼B，则 f ( A ) ∼ f ( B ) , ∣ f ( A ) ∣ ∼ ∣ f ( B ) ∣ f(A) \sim f(B),\left| f(A) \right| \sim \left| f(B) \right| f(A)∼f(B),∣f(A)∣∼∣f(B)∣，其中 f ( A ) f(A) f(A)为关于 n n n阶方阵 A A A的多项式。

3) 若 A A A为可对角化矩阵，则其非零特征值的个数(重根重复计算)＝秩( A A A)

4.实对称矩阵的特征值、特征向量及相似对角阵

(1)相似矩阵：设 A , B A,B A,B为两个 n n n阶方阵，如果存在一个可逆矩阵 P P P，使得 B = P − 1 AP B = P^{- 1}\text{AP} B=P−1AP成立，则称矩阵 A A A与 B B B相似，记为 A ∼ B A \sim B A∼B。

(2)相似矩阵的性质：如果 A ∼ B A \sim B A∼B则有：

1) A T ∼ B T A^{T} \sim B^{T} AT∼BT

2) A − 1 ∼ B − 1 A^{- 1} \sim B^{- 1} A−1∼B−1 （若 A A A， B B B均可逆）

3) A k ∼ B k A^{k} \sim B^{k} Ak∼Bk （ k k k为正整数）

∣ λE − A ∣ = ∣ λE − B ∣ \left| \text{λE} - A \right| = \left| \text{λE} - B \right| ∣λE−A∣=∣λE−B∣，从而 A , B A,B A,B
有相同的特征值

5) ∣ A ∣ = ∣ B ∣ \left| A \right| = \left| B \right| ∣A∣=∣B∣，从而 A , B A,B A,B同时可逆或者不可逆

秩 ( A ) = \left( A \right) = (A)=秩 ( B ) , ∣ λE − A ∣ = ∣ λE − B ∣ \left( B \right),\left| \text{λE} - A \right| = \left| \text{λE} - B \right| (B),∣λE−A∣=∣λE−B∣， A , B A,B A,B不一定相似

二次型

1. n \mathbf{n} n个变量 x 1 , x 2 , ⋯ , x n \mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}\mathbf{,\cdots,}\mathbf{x}_{\mathbf{n}} x1,x2,⋯,xn的二次齐次函数

f ( x 1 , x 2 , ⋯ , x n ) = ∑ i = 1 n ∑ j = 1 n a ij x i y j f(x_{1},x_{2},\cdots,x_{n}) = \sum_{i = 1}^{n}{\sum_{j = 1}^{n}{a_{\text{ij}}x_{i}y_{j}}} f(x1,x2,⋯,xn)=∑i=1n∑j=1naijxiyj，其中 a ij = a ji ( i , j = 1 , 2 , ⋯ , n ) a_{\text{ij}} = a_{\text{ji}}(i,j = 1,2,\cdots,n) aij=aji(i,j=1,2,⋯,n)，称为 n n n元二次型，简称二次型.
若令 x = [ x 1 x 1 ⋮ x n ] , A = [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ ⋯ a n 1 a n 2 ⋯ a nn ] x = \ \begin{bmatrix} x_{1} \\ x_{1} \\ \vdots \\ x_{n} \\ \end{bmatrix},A = \begin{bmatrix} & a_{11}\quad a_{12}\quad\cdots\quad a_{1n} \\ & a_{21}\quad a_{22}\quad\cdots\quad a_{2n} \\ & \quad\cdots\cdots\cdots\cdots\cdots \\ & a_{n1}\quad a_{n2}\quad\cdots\quad a_{\text{nn}} \\ \end{bmatrix} x= x1x1⋮xn ,A= a11a12⋯a1na21a22⋯a2n⋯⋯⋯⋯⋯an1an2⋯ann ,这二次型 f f f可改写成矩阵向量形式 f = x T Ax f = x^{T}\text{Ax} f=xTAx。其中 A A A称为二次型矩阵，因为 a ij = a ji ( i , j = 1 , 2 , ⋯ , n ) a_{\text{ij}} = a_{\text{ji}}(i,j = 1,2,\cdots,n) aij=aji(i,j=1,2,⋯,n)，所以二次型矩阵均为对称矩阵，且二次型与对称矩阵一一对应，并把矩阵 A A A的秩称为二次型的秩。

**2.惯性定理，二次型的标准形和规范形 **

(1) 惯性定理

对于任一二次型，不论选取怎样的合同变换使它化为仅含平方项的标准型，其正负惯性指数与所选变换无关，这就是所谓的惯性定理。

(2) 标准形

二次型 f = ( x 1 , x 2 , ⋯ , x n ) = x T Ax f = \left( x_{1},x_{2},\cdots,x_{n} \right) = x^{T}\text{Ax} f=(x1,x2,⋯,xn)=xTAx经过合同变换 x = Cy x = \text{Cy} x=Cy化为 f = x T Ax = y T C T AC f = x^{T}\text{Ax} = y^{T}C^{T}\text{AC} f=xTAx=yTCTAC

y = ∑ i = 1 r d i y i 2 y = \sum_{i = 1}^{r}{d_{i}y_{i}^{2}} y=∑i=1rdiyi2称为
f ( r ≤ n ) f(r \leq n) f(r≤n)的标准形。在一般的数域内，二次型的标准形不是唯一的，与所作的合同变换有关，但系数不为零的平方项的个数由 r ( A ) r(A) r(A)唯一确定。

(3) 规范形

任一实二次型 f f f都可经过合同变换化为规范形 f = z 1 2 + z 2 2 + ⋯ + z p 2 − z p + 1 2 − ⋯ − z r 2 f = z_{1}^{2} + z_{2}^{2} + \cdots + z_{p}^{2} - z_{p + 1}^{2} - \cdots - z_{r}^{2} f=z12+z22+⋯+zp2−zp+12−⋯−zr2，其中 r r r为 A A A的秩， p p p为正惯性指数， r − p r - p r−p为负惯性指数，且规范型唯一。

**3.用正交变换和配方法化二次型为标准形，二次型及其矩阵的正定性 **

设 A A A正定 ⇒ kA ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow \text{kA}(k > 0),A^{T},A^{- 1},A^{\ast} ⇒kA(k>0),AT,A−1,A∗正定； ∣ A ∣ > 0 |A| > 0 ∣A∣>0, A A A可逆； a ii > 0 a_{\text{ii}} > 0 aii>0，且 ∣ A ii ∣ > 0 |A_{\text{ii}}| > 0 ∣Aii∣>0

A A A， B B B正定 ⇒ A + B \Rightarrow A + B ⇒A+B正定，但 AB \text{AB} AB， BA \text{BA} BA不一定正定

A A A正定 ⇔ f ( x ) = x T Ax > 0 , ∀ x ≠ 0 \Leftrightarrow f(x) = x^{T}\text{Ax} > 0,\forall x \neq 0 ⇔f(x)=xTAx>0,∀x=0

⇔ A \Leftrightarrow A ⇔A的各阶顺序主子式全大于零

⇔ A \Leftrightarrow A ⇔A的所有特征值大于零

⇔ A \Leftrightarrow A ⇔A的正惯性指数为 n n n

⇔ \Leftrightarrow ⇔存在可逆阵 P P P使 A = P T P A = P^{T}P A=PTP

⇔ \Leftrightarrow ⇔存在正交矩阵 Q Q Q，使 Q T AQ = Q − 1 AQ = ( λ 1 ⋱ λ n ) , Q^{T}\text{AQ} = Q^{- 1}\text{AQ} = \begin{pmatrix} \lambda_{1} & & \\ \begin{matrix} & \\ & \\ \end{matrix} & \ddots & \\ & & \lambda_{n} \\ \end{pmatrix}, QTAQ=Q−1AQ= λ1⋱λn ,

其中 λ i > 0 , i = 1 , 2 , ⋯ , n . \lambda_{i} > 0,i = 1,2,\cdots,n. λi>0,i=1,2,⋯,n.正定 ⇒ kA ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow \text{kA}(k > 0),A^{T},A^{- 1},A^{\ast} ⇒kA(k>0),AT,A−1,A∗正定；
∣ A ∣ > 0 , A |A| > 0,A ∣A∣>0,A可逆； a ii > 0 a_{\text{ii}} > 0 aii>0，且 ∣ A ii ∣ > 0 |A_{\text{ii}}| > 0 ∣Aii∣>0 。

概率论和数理统计

随机事件和概率

1.事件的关系与运算

(1) 子事件： A ⊂ B A \subset B A⊂B，若 A A A发生，则 B B B发生。

(2) 相等事件： A = B A = B A=B，即 A ⊂ B A \subset B A⊂B，且 B ⊂ A B \subset A B⊂A 。

(3) 和事件： A ⋃ B A\bigcup B A⋃B（或 A + B A + B A+B）， A A A与 B B B中至少有一个发生。

(4) 差事件： A − B A - B A−B， A A A发生但 B B B不发生。

(5) 积事件： A ⋂ B A\bigcap B A⋂B（或 AB \text{AB} AB）， A A A与 B B B同时发生。

(6) 互斥事件（互不相容）： A ⋂ B A\bigcap B A⋂B= ∅ \varnothing ∅。

(7) 互逆事件（对立事件）：
A ⋂ B = ∅ , A ⋃ B = Ω , A = B ‾ , B = A ‾ A\bigcap B = \varnothing,A\bigcup B = \Omega,A = \overline{B},B = \overline{A} A⋂B=∅,A⋃B=Ω,A=B,B=A
。

2.运算律

(1) 交换律： A ⋃ B = B ⋃ A , A ⋂ B = B ⋂ A A\bigcup B = B\bigcup A,A\bigcap B = B\bigcap A A⋃B=B⋃A,A⋂B=B⋂A

(2) 结合律： ( A ⋃ B ) ⋃ C = A ⋃ ( B ⋃ C ) (A\bigcup B)\bigcup C = A\bigcup(B\bigcup C) (A⋃B)⋃C=A⋃(B⋃C)；
( A ⋂ B ) ⋂ C = A ⋂ ( B ⋂ C ) (A\bigcap B)\bigcap C = A\bigcap(B\bigcap C) (A⋂B)⋂C=A⋂(B⋂C)

(3) 分配律： ( A ⋃ B ) ⋂ C = ( A ⋂ C ) ⋃ ( B ⋂ C ) (A\bigcup B)\bigcap C = (A\bigcap C)\bigcup(B\bigcap C) (A⋃B)⋂C=(A⋂C)⋃(B⋂C)

3.德 . \mathbf{.} .摩根律

A ⋃ B ‾ = A ‾ ⋂ B ‾ \overline{A\bigcup B} = \overline{A}\bigcap\overline{B} A⋃B=A⋂B
A ⋂ B ‾ = A ‾ ⋃ B ‾ \overline{A\bigcap B} = \overline{A}\bigcup\overline{B} A⋂B=A⋃B

4.完全事件组

A 1 A 2 ⋯ A n A_{1}A_{2}\cdots A_{n} A1A2⋯An两两互斥，且和事件为必然事件，即 A i ⋂ A j = ∅ , i ≠ j , ⋃ n i = 1 = Ω A_{i}\bigcap A_{j} = \varnothing,i \neq j,\underset{i = 1}{\bigcup^{n}}\, = \Omega Ai⋂Aj=∅,i=j,i=1⋃n=Ω

5.概率的基本概念

(1) 概率：事件发生的可能性大小的度量，其严格定义如下：

概率 P ( g ) P(g) P(g)为定义在事件集合上的满足下面3个条件的函数：

1)对任何事件 A A A， P ( A ) ≥ 0 P(A) \geq 0 P(A)≥0

2)对必然事件 Ω \Omega Ω， P ( Ω ) = 1 P(\Omega) = 1 P(Ω)=1

3)对 A 1 A 2 ⋯ A n , ⋯ A_{1}A_{2}\cdots A_{n},\cdots A1A2⋯An,⋯
,若 A i A j = ∅ ( i ≠ j ) A_{i}A_{j} = \varnothing(i \neq j) AiAj=∅(i=j)，则： P ( ⋃ ∞ i = 1 A i ) = ∑ i = 1 ∞ P ( A ) . P(\underset{i = 1}{\bigcup^{\infty}}\, A_{i}) = \sum_{i = 1}^{\infty}{P(A).} P(i=1⋃∞Ai)=∑i=1∞P(A).

(2) 概率的基本性质

1) P ( A ‾ ) = 1 − P ( A ) P(\overline{A}) = 1 - P(A) P(A)=1−P(A);

2) P ( A − B ) = P ( A ) − P ( A B ) ; P(A - B) = P(A) - P(AB); P(A−B)=P(A)−P(AB);

3) P ( A ⋃ B ) = P ( A ) + P ( B ) − P ( A B ) P(A\bigcup B) = P(A) + P(B) - P(AB) P(A⋃B)=P(A)+P(B)−P(AB)
特别，当 B ⊂ A B \subset A B⊂A时， P ( A − B ) = P ( A ) − P ( B ) P(A - B) = P(A) - P(B) P(A−B)=P(A)−P(B)且 P ( B ) ≤ P ( A ) P(B) \leq P(A) P(B)≤P(A)；
P ( A ⋃ B ⋃ C ) = P ( A ) + P ( B ) + P ( C ) − P ( A B ) − P ( B C ) − P ( A C ) + P ( A B C ) P(A\bigcup B\bigcup C) = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(AC) + P(ABC) P(A⋃B⋃C)=P(A)+P(B)+P(C)−P(AB)−P(BC)−P(AC)+P(ABC)
4)
若 A 1 , A 2 , ⋯ , A n A_{1},A_{2},\cdots,A_{n} A1,A2,⋯,An两两互斥，则 P ( ⋃ n i = 1 A i ) = ∑ i = 1 n ( P ( A i ) P(\underset{i = 1}{\bigcup^{n}}\, A_{i}) = \sum_{i = 1}^{n}{(P(A_{i})} P(i=1⋃nAi)=∑i=1n(P(Ai)

(3) 古典型概率: 实验的所有结果只有有限个，
且每个结果发生的可能性相同，其概率计算公式： P ( A ) = A P(A) = \frac{A}{} P(A)=A

(4) 几何型概率: 样本空间 Ω \Omega Ω为欧氏空间中的一个区域，
且每个样本点的出现具有等可能性，其概率计算公式： P ( A ) = A ( ) Ω ( ) P(A) = \frac{A()}{\Omega()} P(A)=Ω()A()

6.概率的基本公式

(1) 条件概率: P ( B ∣ A ) = P ( A B ) P ( A ) P(B|A) = \frac{P(AB)}{P(A)} P(B∣A)=P(A)P(AB)
,表示 A A A发生的条件下， B B B发生的概率

(2) 全概率公式：
P ( A ) = ∑ i = 1 n P ( A ∣ B i ) P ( B i ) , B i B j = ∅ , i ≠ j , ⋃ n i = 1 B i = Ω . P(A) = \sum_{i = 1}^{n}{P(A|B_{i})P(B_{i}),B_{i}B_{j}} = \varnothing,i \neq j,\underset{i = 1}{\bigcup^{n}}\, B_{i} = \Omega. P(A)=∑i=1nP(A∣Bi)P(Bi),BiBj=∅,i=j,i=1⋃nBi=Ω.

(3) Bayes公式：

P ( B j ∣ A ) = P ( A ∣ B j ) P ( B j ) ∑ i = 1 n P ( A ∣ B i ) P ( B i ) , j = 1 , 2 , ⋯ , n P(B_{j}|A) = \frac{P(A|B_{j})P(B_{j})}{\sum_{i = 1}^{n}{P(A|B_{i})P(B_{i})}},j = 1,2,\cdots,n P(Bj∣A)=∑i=1nP(A∣Bi)P(Bi)P(A∣Bj)P(Bj),j=1,2,⋯,n

注：上述公式中事件 B i B_{i} Bi的个数可为可列个.

(4)乘法公式：
P ( A 1 A 2 ) = P ( A 1 ) P ( A 2 ∣ A 1 ) = P ( A 2 ) P ( A 1 ∣ A 2 ) P(A_{1}A_{2}) = P(A_{1})P(A_{2}|A_{1}) = P(A_{2})P(A_{1}|A_{2}) P(A1A2)=P(A1)P(A2∣A1)=P(A2)P(A1∣A2)
P ( A 1 A 2 ⋯ A n ) = P ( A 1 ) P ( A 2 ∣ A 1 ) P ( A 3 ∣ A 1 A 2 ) ⋯ P ( A n ∣ A 1 A 2 ⋯ A n − 1 ) P(A_{1}A_{2}\cdots A_{n}) = P(A_{1})P(A_{2}|A_{1})P(A_{3}|A_{1}A_{2})\cdots P(A_{n}|A_{1}A_{2}\cdots A_{n - 1}) P(A1A2⋯An)=P(A1)P(A2∣A1)P(A3∣A1A2)⋯P(An∣A1A2⋯An−1)

7.事件的独立性

(1)
A与B相互独立 ⇔ P ( AB ) = P ( A ) P ( B ) \Leftrightarrow P\left( \text{AB} \right) = P\left( A \right)P\left( B \right) ⇔P(AB)=P(A)P(B)

(2) A，B，C两两独立
⇔ P ( AB ) = P ( A ) P ( B ) ; P ( BC ) = P ( B ) P ( C ) ; \Leftrightarrow P(\text{AB}) = P(A)P(B);P(\text{BC}) = P(B)P(C); ⇔P(AB)=P(A)P(B);P(BC)=P(B)P(C);
P ( AC ) = P ( A ) P ( C ) ; P(\text{AC}) = P(A)P(C); P(AC)=P(A)P(C);

(3) A，B，C相互独立 ⇔ P ( AB ) = P ( A ) P ( B ) ; \Leftrightarrow P(\text{AB}) = P(A)P(B); ⇔P(AB)=P(A)P(B);
P ( BC ) = P ( B ) P ( C ) ; P(\text{BC}) = P(B)P(C); P(BC)=P(B)P(C); P ( AC ) = P ( A ) P ( C ) ; P(\text{AC}) = P(A)P(C); P(AC)=P(A)P(C);
P ( ABC ) = P ( A ) P ( B ) P ( C ) . P(\text{ABC}) = P(A)P(B)P(C). P(ABC)=P(A)P(B)P(C).

8.独立重复试验

将某试验独立重复n次，若每次实验中事件A发生的概率为p，则n次试验中A发生k次的概率为：
$P\left( X = k \right) = C_{n}^{k}p{k}\left( 1 - p \right)^{n - k}\ $。

9.重要公式与结论

(1) P ( A ‾ ) = 1 − P ( A ) P\left( \overline{A} \right) = 1 - P\left( A \right) P(A)=1−P(A)

(2) P ( A ⋃ B ) = P ( A ) + P ( B ) − P ( AB ) P(A\bigcup B) = P(A) + P(B) - P(\text{AB}) P(A⋃B)=P(A)+P(B)−P(AB)

P ( A ⋃ B ⋃ C ) = P ( A ) + P ( B ) + P ( C ) − P ( AB ) − P ( BC ) − P ( AC ) + P ( ABC ) P(A\bigcup B\bigcup C) = P(A) + P(B) + P(C) - P(\text{AB}) - P(\text{BC}) - P(\text{AC}) + P(\text{ABC}) P(A⋃B⋃C)=P(A)+P(B)+P(C)−P(AB)−P(BC)−P(AC)+P(ABC)

(3)
P ( A − B ) = P ( A ) − P ( AB ) P\left( A - B \right) = P\left( A \right) - P\left( \text{AB} \right) P(A−B)=P(A)−P(AB)

(4)
P ( A B ‾ ) = P ( A ) − P ( AB ) , P ( A ) = P ( AB ) + P ( A B ‾ ) , P(A\overline{B}) = P(A) - P(\text{AB}),P(A) = P(\text{AB}) + P(A\overline{B}), P(AB)=P(A)−P(AB),P(A)=P(AB)+P(AB),
P ( A ⋃ B ) = P ( A ) + P ( A ‾ B ) = P ( AB ) + P ( A B ‾ ) + P ( A ‾ B ) P(A\bigcup B) = P(A) + P(\overline{A}B) = P(\text{AB}) + P(A\overline{B}) + P(\overline{A}B) P(A⋃B)=P(A)+P(AB)=P(AB)+P(AB)+P(AB)

(5) 条件概率 P ( ∣ B ) P(|B) P(∣B)满足概率的所有性质，

例如：. P ( A ‾ 1 ∣ B ) = 1 − P ( A 1 ∣ B ) P({\overline{A}}_{1}|B) = 1 - P(A_{1}|B) P(A1∣B)=1−P(A1∣B)
P ( A 1 ⋃ A 2 ∣ B ) = P ( A 1 ∣ B ) + P ( A 2 ∣ B ) − P ( A 1 A 2 ∣ B ) P(A_{1}\bigcup A_{2}|B) = P(A_{1}|B) + P(A_{2}|B) - P(A_{1}A_{2}|B) P(A1⋃A2∣B)=P(A1∣B)+P(A2∣B)−P(A1A2∣B)
P ( A 1 A 2 ∣ B ) = P ( A 1 ∣ B ) P ( A 2 ∣ A 1 B ) P(A_{1}A_{2}|B) = P(A_{1}|B)P(A_{2}|A_{1}B) P(A1A2∣B)=P(A1∣B)P(A2∣A1B)

(6)
若 A 1 , A 2 , ⋯ , A n A_{1},A_{2},\cdots,A_{n} A1,A2,⋯,An相互独立，则 P ( ⋂ i = 1 n A i ) = ∏ i = 1 n P ( A i ) , P(\bigcap_{i = 1}^{n}A_{i}) = \prod_{i = 1}^{n}{P(A_{i})}, P(⋂i=1nAi)=∏i=1nP(Ai),
P ( ⋃ i = 1 n A i ) = ∏ i = 1 n ( 1 − P ( A i ) ) P(\bigcup_{i = 1}^{n}A_{i}) = \prod_{i = 1}^{n}{(1 - P(A_{i}))} P(⋃i=1nAi)=∏i=1n(1−P(Ai))

(7) 互斥、互逆与独立性之间的关系：
A与B互逆 ⇒ \Rightarrow ⇒A与B互斥，但反之不成立，A与B互
斥（或互逆）且均非零概率事件 ⇒ \Rightarrow ⇒A与B不独立.

(8)
若 A 1 , A 2 , ⋯ , A m , B 1 , B 2 , ⋯ , B n A_{1},A_{2},\cdots,A_{m},B_{1},B_{2},\cdots,B_{n} A1,A2,⋯,Am,B1,B2,⋯,Bn相互独立，则 f ( A 1 , A 2 , ⋯ , A m ) f(A_{1},A_{2},\cdots,A_{m}) f(A1,A2,⋯,Am)与
g ( B 1 , B 2 , ⋯ , B n ) g(B_{1},B_{2},\cdots,B_{n}) g(B1,B2,⋯,Bn)也相互独立，其中 f ( ) , g ( ) f(),g() f(),g()分别表示对相应事件做任意事件运算后所得的事件，另外，概率为1（或0）的事件与任何事件相互独立.

随机变量及其概率分布

1.随机变量及概率分布

取值带有随机性的变量，严格地说是定义在样本空间上，取值于实数的函数称为随机变量，概率分布通常指分布函数或分布律

2.分布函数的概念与性质

定义： F ( x ) = P ( X ≤ x ) , − ∞ < x < + ∞ F(x) = P(X \leq x), - \infty < x < + \infty F(x)=P(X≤x),−∞<x<+∞

性质：(1) 0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 1 0≤F(x)≤1 (2) F ( x ) F(x) F(x)单调不减

(3)右连续 F ( x + 0 ) = F ( x ) F(x + 0) = F(x) F(x+0)=F(x) (4) F ( − ∞ ) = 0 , F ( + ∞ ) = 1 F( - \infty) = 0,F( + \infty) = 1 F(−∞)=0,F(+∞)=1

3.离散型随机变量的概率分布

P ( X = x i ) = p i , i = 1 , 2 , ⋯ , n , ⋯ p i ≥ 0 , ∑ i = 1 ∞ p i = 1 P(X = x_{i}) = p_{i},i = 1,2,\cdots,n,\cdots\quad\quad p_{i} \geq 0,\sum_{i = 1}^{\infty}p_{i} = 1 P(X=xi)=pi,i=1,2,⋯,n,⋯pi≥0,∑i=1∞pi=1

4.连续型随机变量的概率密度

概率密度 f ( x ) ; f(x); f(x);非负可积，且:(1) f ( x ) ≥ 0 , f(x) \geq 0, f(x)≥0,
(2) ∫ − ∞ + ∞ f ( x ) dx = 1 \int_{- \infty}^{+ \infty}{f(x)\text{dx} = 1} ∫−∞+∞f(x)dx=1
(3) x x x为 f ( x ) f(x) f(x)的连续点，则:

f ( x ) = F ′ ( x ) f(x) = F'(x) f(x)=F′(x)分布函数 F ( x ) = ∫ − ∞ x f ( t ) dt F(x) = \int_{- \infty}^{x}{f(t)\text{dt}} F(x)=∫−∞xf(t)dt

5.常见分布

(1) 0-1分布: P ( X = k ) = p k ( 1 − p ) 1 − k , k = 0 , 1 P(X = k) = p^{k}{(1 - p)}^{1 - k},k = 0,1 P(X=k)=pk(1−p)1−k,k=0,1

(2) 二项分布: B ( n , p ) B(n,p) B(n,p)：
P ( X = k ) = C n k p k ( 1 − p ) n − k , k = 0 , 1 , ⋯ , n P(X = k) = C_{n}^{k}p^{k}{(1 - p)}^{n - k},k = 0,1,\cdots,n P(X=k)=Cnkpk(1−p)n−k,k=0,1,⋯,n

(3) Poisson分布: p ( λ ) p(\lambda) p(λ)：
P ( X = k ) = λ k k ! e − λ , λ > 0 , k = 0 , 1 , 2 ⋯ P(X = k) = \frac{\lambda^{k}}{k!}e^{- \lambda},\lambda > 0,k = 0,1,2\cdots P(X=k)=k!λke−λ,λ>0,k=0,1,2⋯

(4) 均匀分布 U ( a , b ) U(a,b) U(a,b)：$f(x) = \left{ \begin{matrix}
& \frac{1}{b - a},a < x < b \
& 0, \
\end{matrix} \right.\ $

(5) 正态分布: N ( μ , σ 2 ) : N(\mu,\sigma^{2}): N(μ,σ2):
φ ( x ) = 1 2 π σ e − ( x − μ ) 2 2 σ 2 , σ > 0 , − ∞ < x < + ∞ \varphi(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{{(x - \mu)}^{2}}{2\sigma^{2}}},\sigma > 0, - \infty < x < + \infty φ(x)=2π σ1e−2σ2(x−μ)2,σ>0,−∞<x<+∞

(6)指数分布:$E(\lambda):f(x) = \left{ \begin{matrix}
& \lambda e^{- \text{λx}},x > 0,\lambda > 0 \
& 0, \
\end{matrix} \right.\ $

(7)几何分布: G ( p ) : P ( X = k ) = ( 1 − p ) k − 1 p , 0 < p < 1 , k = 1 , 2 , ⋯ . G(p):P(X = k) = {(1 - p)}^{k - 1}p,0 < p < 1,k = 1,2,\cdots. G(p):P(X=k)=(1−p)k−1p,0<p<1,k=1,2,⋯.

(8)超几何分布:
H ( N , M , n ) : P ( X = k ) = C M k C N − M n − k C N n , k = 0 , 1 , ⋯ , m i n ( n , M ) H(N,M,n):P(X = k) = \frac{C_{M}^{k}C_{N - M}^{n - k}}{C_{N}^{n}},k = 0,1,\cdots,min(n,M) H(N,M,n):P(X=k)=CNnCMkCN−Mn−k,k=0,1,⋯,min(n,M)

6.随机变量函数的概率分布

(1)离散型： P ( X = x 1 ) = p i , Y = g ( X ) P(X = x_{1}) = p_{i},Y = g(X) P(X=x1)=pi,Y=g(X)

则: P ( Y = y j ) = ∑ g ( x i ) = y i P ( X = x i ) P(Y = y_{j}) = \sum_{g(x_{i}) = y_{i}}^{}{P(X = x_{i})} P(Y=yj)=∑g(xi)=yiP(X=xi)

(2)连续型： X ~ f X ( x ) , Y = g ( x ) X\tilde{\ }f_{X}(x),Y = g(x) X ~fX(x),Y=g(x)

则: F y ( y ) = P ( Y ≤ y ) = P ( g ( X ) ≤ y ) = ∫ g ( x ) ≤ y f x ( x ) d x F_{y}(y) = P(Y \leq y) = P(g(X) \leq y) = \int_{g(x) \leq y}^{}{f_{x}(x)dx} Fy(y)=P(Y≤y)=P(g(X)≤y)=∫g(x)≤yfx(x)dx，
f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F'_{Y}(y) fY(y)=FY′(y)

7.重要公式与结论

(1)
X ∼ N ( 0 , 1 ) ⇒ φ ( 0 ) = 1 2 π , Φ ( 0 ) = 1 2 , X\sim N(0,1) \Rightarrow \varphi(0) = \frac{1}{\sqrt{2\pi}},\Phi(0) = \frac{1}{2}, X∼N(0,1)⇒φ(0)=2π 1,Φ(0)=21,
Φ ( − a ) = P ( X ≤ − a ) = 1 − Φ ( a ) \Phi( - a) = P(X \leq - a) = 1 - \Phi(a) Φ(−a)=P(X≤−a)=1−Φ(a)

(2)
X ∼ N ( μ , σ 2 ) ⇒ X − μ σ ∼ N ( 0 , 1 ) , P ( X ≤ a ) = Φ ( a − μ σ ) X\sim N\left( \mu,\sigma^{2} \right) \Rightarrow \frac{X - \mu}{\sigma}\sim N\left( 0,1 \right),P(X \leq a) = \Phi(\frac{a - \mu}{\sigma}) X∼N(μ,σ2)⇒σX−μ∼N(0,1),P(X≤a)=Φ(σa−μ)

(3) X ∼ E ( λ ) ⇒ P ( X > s + t ∣ X > s ) = P ( X > t ) X\sim E(\lambda) \Rightarrow P(X > s + t|X > s) = P(X > t) X∼E(λ)⇒P(X>s+t∣X>s)=P(X>t)

(4) X ∼ G ( p ) ⇒ P ( X = m + k ∣ X > m ) = P ( X = k ) X\sim G(p) \Rightarrow P(X = m + k|X > m) = P(X = k) X∼G(p)⇒P(X=m+k∣X>m)=P(X=k)

(5)
离散型随机变量的分布函数为阶梯间断函数；连续型随机变量的分布函数为连续函数，但不一定为处处可导函数。

(6) 存在既非离散也非连续型随机变量。

多维随机变量及其分布

**1.二维随机变量及其联合分布 **

由两个随机变量构成的随机向量 ( X , Y ) (X,Y) (X,Y)，
联合分布为 F ( x , y ) = P ( X ≤ x , Y ≤ y ) F(x,y) = P(X \leq x,Y \leq y) F(x,y)=P(X≤x,Y≤y)

2.二维离散型随机变量的分布

(1) 联合概率分布律
P { X = x i , Y = y j } = p ij ; i , j = 1 , 2 , ⋯ P\{ X = x_{i},Y = y_{j}\} = p_{\text{ij}};i,j = 1,2,\cdots P{X=xi,Y=yj}=pij;i,j=1,2,⋯

(2) 边缘分布律
p i ⋅ = ∑ j = 1 ∞ p ij , i = 1 , 2 , ⋯ p_{i \cdot} = \sum_{j = 1}^{\infty}p_{\text{ij}},i = 1,2,\cdots pi⋅=∑j=1∞pij,i=1,2,⋯
p ⋅ j = ∑ i ∞ p ij , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{\text{ij}},j = 1,2,\cdots p⋅j=∑i∞pij,j=1,2,⋯

(3) 条件分布律
P { X = x i ∣ Y = y j } = p ij p ⋅ j P\{ X = x_{i}|Y = y_{j}\} = \frac{p_{\text{ij}}}{p_{\cdot j}} P{X=xi∣Y=yj}=p⋅jpij
P { Y = y j ∣ X = x i } = p ij p i ⋅ P\{ Y = y_{j}|X = x_{i}\} = \frac{p_{\text{ij}}}{p_{i \cdot}} P{Y=yj∣X=xi}=pi⋅pij

**3. 二维连续性随机变量的密度 **

(1) 联合概率密度 f ( x , y ) : f(x,y): f(x,y):

1) f ( x , y ) ≥ 0 f(x,y) \geq 0 f(x,y)≥0 2)
∫ − ∞ + ∞ ∫ − ∞ + ∞ f ( x , y ) d x d y = 1 \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{f(x,y)dxdy}} = 1 ∫−∞+∞∫−∞+∞f(x,y)dxdy=1

(2)
分布函数： F ( x , y ) = ∫ − ∞ x ∫ − ∞ y f ( u , v ) d u d v F(x,y) = \int_{- \infty}^{x}{\int_{- \infty}^{y}{f(u,v)dudv}} F(x,y)=∫−∞x∫−∞yf(u,v)dudv

(3) 边缘概率密度：
f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) dy f_{X}\left( x \right) = \int_{- \infty}^{+ \infty}{f\left( x,y \right)\text{dy}} fX(x)=∫−∞+∞f(x,y)dy
f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx} fY(y)=∫−∞+∞f(x,y)dx

(4)
条件概率密度： f X ∣ Y ( x | y ) = f ( x , y ) f Y ( y ) f_{X|Y}\left( x \middle| y \right) = \frac{f\left( x,y \right)}{f_{Y}\left( y \right)} fX∣Y(x∣y)=fY(y)f(x,y)
f Y ∣ X ( y ∣ x ) = f ( x , y ) f X ( x ) f_{Y|X}(y|x) = \frac{f(x,y)}{f_{X}(x)} fY∣X(y∣x)=fX(x)f(x,y)

4.常见二维随机变量的联合分布

(1) 二维均匀分布： ( x , y ) ∼ U ( D ) (x,y) \sim U(D) (x,y)∼U(D) ,$f(x,y) = \left{ \begin{matrix}
& \frac{1}{S(D)},(x,y) \in D \
& 0,\ \ \
\end{matrix} \right.\ $

(2)
二维正态分布：( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) X,Y)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) X,Y)∼N(μ1,μ2,σ12,σ22,ρ)

f ( x , y ) = 1 2 π σ 1 σ 2 1 − ρ 2 . exp ⁡ { − 1 2 ( 1 − ρ 2 ) [ ( x − μ 1 ) 2 σ 1 2 − 2 ρ ( x − μ 1 ) ( y − μ 2 ) σ 1 σ 2 + ( y − μ 2 ) 2 σ 2 2 ] } f(x,y) = \frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1 - \rho^{2}}}.\exp\left\{ \frac{- 1}{2(1 - \rho^{2})}\lbrack\frac{{(x - \mu_{1})}^{2}}{\sigma_{1}^{2}} - 2\rho\frac{(x - \mu_{1})(y - \mu_{2})}{\sigma_{1}\sigma_{2}} + \frac{{(y - \mu_{2})}^{2}}{\sigma_{2}^{2}}\rbrack \right\} f(x,y)=2πσ1σ21−ρ2 1.exp{2(1−ρ2)−1[σ12(x−μ1)2−2ρσ1σ2(x−μ1)(y−μ2)+σ22(y−μ2)2]}

**5.随机变量的独立性和相关性 **

X X X和 Y Y Y的相互独立: ⇔ F ( x , y ) = F X ( x ) F Y ( y ) \Leftrightarrow F\left( x,y \right) = F_{X}\left( x \right)F_{Y}\left( y \right) ⇔F(x,y)=FX(x)FY(y):

⇔ p ij = p i ⋅ ⋅ p ⋅ j \Leftrightarrow p_{\text{ij}} = p_{i \cdot} \cdot p_{\cdot j} ⇔pij=pi⋅⋅p⋅j（离散型）
⇔ f ( x , y ) = f X ( x ) f Y ( y ) \Leftrightarrow f\left( x,y \right) = f_{X}\left( x \right)f_{Y}\left( y \right) ⇔f(x,y)=fX(x)fY(y)（连续型）

X X X和 Y Y Y的相关性：

相关系数 ρ XY = 0 \rho_{\text{XY}} = 0 ρXY=0时，称 X X X和 Y Y Y不相关，否则称 X X X和 Y Y Y相关

6.两个随机变量简单函数的概率分布

离散型：
P ( X = x i , Y = y i ) = p ij , Z = g ( X , Y ) P\left( X = x_{i},Y = y_{i} \right) = p_{\text{ij}},Z = g\left( X,Y \right) P(X=xi,Y=yi)=pij,Z=g(X,Y)
则：

P ( Z = z k ) = P { g ( X , Y ) = z k } = ∑ g ( x i , y i ) = z k P ( X = x i , Y = y j ) P(Z = z_{k}) = P\left\{ g\left( X,Y \right) = z_{k} \right\} = \sum_{g\left( x_{i},y_{i} \right) = z_{k}}^{}{P\left( X = x_{i},Y = y_{j} \right)} P(Z=zk)=P{g(X,Y)=zk}=∑g(xi,yi)=zkP(X=xi,Y=yj)

连续型：
( X , Y ) ∼ f ( x , y ) , Z = g ( X , Y ) \left( X,Y \right) \sim f\left( x,y \right),Z = g\left( X,Y \right) (X,Y)∼f(x,y),Z=g(X,Y)
则：

F z ( z ) = P { g ( X , Y ) ≤ z } = ∬ g ( x , y ) ≤ z f ( x , y ) d x d y F_{z}\left( z \right) = P\left\{ g\left( X,Y \right) \leq z \right\} = \iint_{g(x,y) \leq z}^{}{f(x,y)dxdy} Fz(z)=P{g(X,Y)≤z}=∬g(x,y)≤zf(x,y)dxdy， f z ( z ) = F z ′ ( z ) f_{z}(z) = F'_{z}(z) fz(z)=Fz′(z)

**7.重要公式与结论 **

(1) 边缘密度公式： f X ( x ) = ∫ − ∞ + ∞ f ( x , y ) d y , f_{X}(x) = \int_{- \infty}^{+ \infty}{f(x,y)dy,} fX(x)=∫−∞+∞f(x,y)dy,
f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx} fY(y)=∫−∞+∞f(x,y)dx

(2)
P { ( X , Y ) ∈ D } = ∬ D f ( x , y ) dxdy P\left\{ \left( X,Y \right) \in D \right\} = \iint_{D}^{}{f\left( x,y \right)\text{dxdy}} P{(X,Y)∈D}=∬Df(x,y)dxdy

(3)
若 ( X , Y ) (X,Y) (X,Y)服从二维正态分布 N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , ρ ) N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho) N(μ1,μ2,σ12,σ22,ρ)
则有：

X ∼ N ( μ 1 , σ 1 2 ) , Y ∼ N ( μ 2 , σ 2 2 ) . X\sim N\left( \mu_{1},\sigma_{1}^{2} \right),Y\sim N(\mu_{2},\sigma_{2}^{2}). X∼N(μ1,σ12),Y∼N(μ2,σ22).

2) X X X与 Y Y Y相互独立 ⇔ ρ = 0 \Leftrightarrow \rho = 0 ⇔ρ=0，即 X X X与 Y Y Y不相关。

C 1 X + C 2 Y ∼ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 + C 2 2 σ 2 2 + 2 C 1 C 2 σ 1 σ 2 ρ ) C_{1}X + C_{2}Y\sim N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} + C_{2}^{2}\sigma_{2}^{2} + 2C_{1}C_{2}\sigma_{1}\sigma_{2}\rho) C1X+C2Y∼N(C1μ1+C2μ2,C12σ12+C22σ22+2C1C2σ1σ2ρ)

4) X \text{\ X} X关于Y=y的条件分布为：
N ( μ 1 + ρ σ 1 σ 2 ( y − μ 2 ) , σ 1 2 ( 1 − ρ 2 ) ) N(\mu_{1} + \rho\frac{\sigma_{1}}{\sigma_{2}}(y - \mu_{2}),\sigma_{1}^{2}(1 - \rho^{2})) N(μ1+ρσ2σ1(y−μ2),σ12(1−ρ2))

5) Y Y Y关于 X = x X = x X=x的条件分布为：
N ( μ 2 + ρ σ 2 σ 1 ( x − μ 1 ) , σ 2 2 ( 1 − ρ 2 ) ) N(\mu_{2} + \rho\frac{\sigma_{2}}{\sigma_{1}}(x - \mu_{1}),\sigma_{2}^{2}(1 - \rho^{2})) N(μ2+ρσ1σ2(x−μ1),σ22(1−ρ2))

(4)
若 X X X与 Y Y Y独立，且分别服从 N ( μ 1 , σ 1 2 ) , N ( μ 1 , σ 2 2 ) , N(\mu_{1},\sigma_{1}^{2}),N(\mu_{1},\sigma_{2}^{2}), N(μ1,σ12),N(μ1,σ22),
则：

( X , Y ) ∼ N ( μ 1 , μ 2 , σ 1 2 , σ 2 2 , 0 ) , \left( X,Y \right)\sim N(\mu_{1},\mu_{2},\sigma_{1}^{2},\sigma_{2}^{2},0), (X,Y)∼N(μ1,μ2,σ12,σ22,0),
C 1 X + C 2 Y ~ N ( C 1 μ 1 + C 2 μ 2 , C 1 2 σ 1 2 + C 2 2 σ 2 2 ) . C_{1}X + C_{2}Y\tilde{\ }N(C_{1}\mu_{1} + C_{2}\mu_{2},C_{1}^{2}\sigma_{1}^{2} + C_{2}^{2}\sigma_{2}^{2}). C1X+C2Y ~N(C1μ1+C2μ2,C12σ12+C22σ22).

(5)
若 X X X与 Y Y Y相互独立， f ( x ) f\left( x \right) f(x)和 g ( x ) g\left( x \right) g(x)为连续函数，
则 f ( X ) f\left( X \right) f(X)和 g ( Y ) g(Y) g(Y)也相互独立。

随机变量的数字特征

**1.数学期望 **

离散型： P { X = x i } = p i , E ( X ) = ∑ i x i p i P\left\{ X = x_{i} \right\} = p_{i},E(X) = \sum_{i}^{}{x_{i}p_{i}} P{X=xi}=pi,E(X)=∑ixipi；

连续型： X ∼ f ( x ) , E ( X ) = ∫ − ∞ + ∞ x f ( x ) d x X\sim f(x),E(X) = \int_{- \infty}^{+ \infty}{xf(x)dx} X∼f(x),E(X)=∫−∞+∞xf(x)dx

性质：

(1) E ( C ) = C , E [ E ( X ) ] = E ( X ) E(C) = C,E\lbrack E(X)\rbrack = E(X) E(C)=C,E[E(X)]=E(X)

(2) E ( C 1 X + C 2 Y ) = C 1 E ( X ) + C 2 E ( Y ) E(C_{1}X + C_{2}Y) = C_{1}E(X) + C_{2}E(Y) E(C1X+C2Y)=C1E(X)+C2E(Y)

(3) 若X和Y独立，则 E ( X Y ) = E ( X ) E ( Y ) E(XY) = E(X)E(Y) E(XY)=E(X)E(Y)
(4) [ E ( X Y ) ] 2 ≤ E ( X 2 ) E ( Y 2 ) \left\lbrack E(XY) \right\rbrack^{2} \leq E(X^{2})E(Y^{2}) [E(XY)]2≤E(X2)E(Y2)

2.方差： D ( X ) = E [ X − E ( X ) ] 2 = E ( X 2 ) − [ E ( X ) ] 2 D(X) = E\left\lbrack X - E(X) \right\rbrack^{2} = E(X^{2}) - \left\lbrack E(X) \right\rbrack^{2} D(X)=E[X−E(X)]2=E(X2)−[E(X)]2

3.标准差： D ( X ) \sqrt{D(X)} D(X) ，

4.离散型： D ( X ) = ∑ i [ x i − E ( X ) ] 2 p i D(X) = \sum_{i}^{}{\left\lbrack x_{i} - E(X) \right\rbrack^{2}p_{i}} D(X)=∑i[xi−E(X)]2pi

5.连续型： D ( X ) = ∫ − ∞ + ∞ [ x − E ( X ) ] 2 f ( x ) d x D(X) = {\int_{- \infty}^{+ \infty}\left\lbrack x - E(X) \right\rbrack}^{2}f(x)dx D(X)=∫−∞+∞[x−E(X)]2f(x)dx

性质：

(1) D ( C ) = 0 , D [ E ( X ) ] = 0 , D [ D ( X ) ] = 0 \ D(C) = 0,D\lbrack E(X)\rbrack = 0,D\lbrack D(X)\rbrack = 0 D(C)=0,D[E(X)]=0,D[D(X)]=0

(2) X \ X X与 Y Y Y相互独立，则 D ( X ± Y ) = D ( X ) + D ( Y ) D(X \pm Y) = D(X) + D(Y) D(X±Y)=D(X)+D(Y)

(3) D ( C 1 X + C 2 ) = C 1 2 D ( X ) \ D\left( C_{1}X + C_{2} \right) = C_{1}^{2}D\left( X \right) D(C1X+C2)=C12D(X)

(4) 一般有
D ( X ± Y ) = D ( X ) + D ( Y ) ± 2 C o v ( X , Y ) = D ( X ) + D ( Y ) ± 2 ρ D ( X ) D ( Y ) D(X \pm Y) = D(X) + D(Y) \pm 2Cov(X,Y) = D(X) + D(Y) \pm 2\rho\sqrt{D(X)}\sqrt{D(Y)} D(X±Y)=D(X)+D(Y)±2Cov(X,Y)=D(X)+D(Y)±2ρD(X) D(Y)

(5) D ( X ) < E ( X − C ) 2 , C ≠ E ( X ) \ D\left( X \right) < E\left( X - C \right)^{2},C \neq E\left( X \right) D(X)<E(X−C)2,C=E(X)

(6) D ( X ) = 0 ⇔ P { X = C } = 1 \ D(X) = 0 \Leftrightarrow P\left\{ X = C \right\} = 1 D(X)=0⇔P{X=C}=1

**6.随机变量函数的数学期望 **

(1) 对于函数 Y = g ( x ) Y = g(x) Y=g(x)

X X X为离散型： P { X = x i } = p i , E ( Y ) = ∑ i g ( x i ) p i P\{ X = x_{i}\} = p_{i},E(Y) = \sum_{i}^{}{g(x_{i})p_{i}} P{X=xi}=pi,E(Y)=∑ig(xi)pi；

X X X为连续型： X ∼ f ( x ) , E ( Y ) = ∫ − ∞ + ∞ g ( x ) f ( x ) d x X\sim f(x),E(Y) = \int_{- \infty}^{+ \infty}{g(x)f(x)dx} X∼f(x),E(Y)=∫−∞+∞g(x)f(x)dx

(2)
Z = g ( X , Y ) Z = g(X,Y) Z=g(X,Y); ( X , Y ) ∼ P { X = x i , Y = y j } = p ij \left( X,Y \right)\sim P\{ X = x_{i},Y = y_{j}\} = p_{\text{ij}} (X,Y)∼P{X=xi,Y=yj}=pij;
E ( Z ) = ∑ i ∑ j g ( x i , y j ) p ij E(Z) = \sum_{i}^{}{\sum_{j}^{}{g(x_{i},y_{j})p_{\text{ij}}}} E(Z)=∑i∑jg(xi,yj)pij
( X , Y ) ∼ f ( x , y ) \left( X,Y \right)\sim f(x,y) (X,Y)∼f(x,y); E ( Z ) = ∫ − ∞ + ∞ ∫ − ∞ + ∞ g ( x , y ) f ( x , y ) d x d y E(Z) = \int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{g(x,y)f(x,y)dxdy}} E(Z)=∫−∞+∞∫−∞+∞g(x,y)f(x,y)dxdy

7.协方差
C o v ( X , Y ) = E [ ( X − E ( X ) ( Y − E ( Y ) ) ] Cov(X,Y) = E\left\lbrack (X - E(X)(Y - E(Y)) \right\rbrack Cov(X,Y)=E[(X−E(X)(Y−E(Y))]

8.相关系数
ρ XY = C o v ( X , Y ) D ( X ) D ( Y ) \rho_{\text{XY}} = \frac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}} ρXY=D(X) D(Y) Cov(X,Y), k k k阶原点矩
E ( X k ) E(X^{k}) E(Xk); k k k阶中心矩
E { [ X − E ( X ) ] k } E\left\{ {\lbrack X - E(X)\rbrack}^{k} \right\} E{[X−E(X)]k}

性质：

(1) C o v ( X , Y ) = C o v ( Y , X ) \ Cov(X,Y) = Cov(Y,X) Cov(X,Y)=Cov(Y,X)

(2) C o v ( a X , b Y ) = a b C o v ( Y , X ) \ Cov(aX,bY) = abCov(Y,X) Cov(aX,bY)=abCov(Y,X)

(3) C o v ( X 1 + X 2 , Y ) = C o v ( X 1 , Y ) + C o v ( X 2 , Y ) \ Cov(X_{1} + X_{2},Y) = Cov(X_{1},Y) + Cov(X_{2},Y) Cov(X1+X2,Y)=Cov(X1,Y)+Cov(X2,Y)

(4) ∣ ρ ( X , Y ) ∣ ≤ 1 \ \left| \rho\left( X,Y \right) \right| \leq 1 ∣ρ(X,Y)∣≤1

(5) ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \ \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1⇔P(Y=aX+b)=1
，其中 a > 0 a > 0 a>0

ρ ( X , Y ) = − 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=−1⇔P(Y=aX+b)=1
，其中 a < 0 a < 0 a<0

9.重要公式与结论

(1) D ( X ) = E ( X 2 ) − E 2 ( X ) \ D(X) = E(X^{2}) - E^{2}(X) D(X)=E(X2)−E2(X)

(2) C o v ( X , Y ) = E ( X Y ) − E ( X ) E ( Y ) \ Cov(X,Y) = E(XY) - E(X)E(Y) Cov(X,Y)=E(XY)−E(X)E(Y)

(3) ∣ ρ ( X , Y ) ∣ ≤ 1 , \left| \rho\left( X,Y \right) \right| \leq 1, ∣ρ(X,Y)∣≤1,且
ρ ( X , Y ) = 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=1⇔P(Y=aX+b)=1，其中 a > 0 a > 0 a>0

ρ ( X , Y ) = − 1 ⇔ P ( Y = a X + b ) = 1 \rho\left( X,Y \right) = - 1 \Leftrightarrow P\left( Y = aX + b \right) = 1 ρ(X,Y)=−1⇔P(Y=aX+b)=1，其中 a < 0 a < 0 a<0

(4) 下面5个条件互为充要条件：

ρ ( X , Y ) = 0 \rho(X,Y) = 0 ρ(X,Y)=0 ⇔ C o v ( X , Y ) = 0 \Leftrightarrow Cov(X,Y) = 0 ⇔Cov(X,Y)=0
⇔ E ( X , Y ) = E ( X ) E ( Y ) \Leftrightarrow E(X,Y) = E(X)E(Y) ⇔E(X,Y)=E(X)E(Y)
⇔ D ( X + Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X + Y) = D(X) + D(Y) ⇔D(X+Y)=D(X)+D(Y)
⇔ D ( X − Y ) = D ( X ) + D ( Y ) \Leftrightarrow D(X - Y) = D(X) + D(Y) ⇔D(X−Y)=D(X)+D(Y)

注： X X X与 Y Y Y独立为上述5个条件中任何一个成立的充分条件，但非必要条件。

数理统计的基本概念

1.基本概念

总体：研究对象的全体，它是一个随机变量，用 X X X表示。

个体：组成总体的每个基本元素。

简单随机样本：来自总体 X X X的 n n n个相互独立且与总体同分布的随机变量 X 1 , X 2 ⋯ , X n X_{1},X_{2}\cdots,X_{n} X1,X2⋯,Xn，称为容量为 n n n的简单随机样本，简称样本。

统计量：设 X 1 , X 2 ⋯ , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2⋯,Xn,是来自总体 X X X的一个样本， g ( X 1 , X 2 ⋯ , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2⋯,Xn)）是样本的连续函数，且 g ( ) g() g()中不含任何未知参数，则称 g ( X 1 , X 2 ⋯ , X n ) g(X_{1},X_{2}\cdots,X_{n}) g(X1,X2⋯,Xn)为统计量

样本均值： X ‾ = 1 n ∑ i = 1 n X i \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i} X=n1∑i=1nXi

样本方差： S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{2} S2=n−11∑i=1n(Xi−X)2

样本矩：样本 k k k阶原点矩： A k = 1 n ∑ i = 1 n X i k , k = 1 , 2 , ⋯ A_{k} = \frac{1}{n}\sum_{i = 1}^{n}X_{i}^{k},k = 1,2,\cdots Ak=n1∑i=1nXik,k=1,2,⋯

样本 k k k阶中心矩： B k = 1 n ∑ i = 1 n ( X i − X ‾ ) k , k = 1 , 2 , ⋯ B_{k} = \frac{1}{n}\sum_{i = 1}^{n}{(X_{i} - \overline{X})}^{k},k = 1,2,\cdots Bk=n1∑i=1n(Xi−X)k,k=1,2,⋯

2.分布

χ 2 \chi^{2} χ2分布： χ 2 = X 1 2 + X 2 2 + ⋯ + X n 2 ∼ χ 2 ( n ) \chi^{2} = X_{1}^{2} + X_{2}^{2} + \cdots + X_{n}^{2}\sim\chi^{2}(n) χ2=X12+X22+⋯+Xn2∼χ2(n)，其中 X 1 , X 2 ⋯ , X n , X_{1},X_{2}\cdots,X_{n}, X1,X2⋯,Xn,相互独立，且同服从 N ( 0 , 1 ) N(0,1) N(0,1)

t t t分布： T = X Y / n ∼ t ( n ) T = \frac{X}{\sqrt{Y/n}}\sim t(n) T=Y/n X∼t(n)
，其中 X ∼ N ( 0 , 1 ) , Y ∼ χ 2 ( n ) , X\sim N\left( 0,1 \right),Y\sim\chi^{2}(n), X∼N(0,1),Y∼χ2(n),且 X X X， Y Y Y 相互独立。

F分布： F = X / n 1 Y / n 2 ∼ F ( n 1 , n 2 ) F = \frac{X/n_{1}}{Y/n_{2}}\sim F(n_{1},n_{2}) F=Y/n2X/n1∼F(n1,n2)，其中 X ∼ χ 2 ( n 1 ) , Y ∼ χ 2 ( n 2 ) , X\sim\chi^{2}\left( n_{1} \right),Y\sim\chi^{2}(n_{2}), X∼χ2(n1),Y∼χ2(n2),且 X X X， Y Y Y相互独立。

分位数：若 P ( X ≤ x α ) = α , P(X \leq x_{\alpha}) = \alpha, P(X≤xα)=α,则称 x α x_{\alpha} xα为 X X X的 α \alpha α分位数

**3.正态总体的常用样本分布 **

(1) 设 X 1 , X 2 ⋯ , X n X_{1},X_{2}\cdots,X_{n} X1,X2⋯,Xn为来自正态总体 N ( μ , σ 2 ) N(\mu,\sigma^{2}) N(μ,σ2)的样本，

X ‾ = 1 n ∑ i = 1 n X i , S 2 = 1 n − 1 ∑ i = 1 n ( X i − X ‾ ) 2 , \overline{X} = \frac{1}{n}\sum_{i = 1}^{n}X_{i},S^{2} = \frac{1}{n - 1}\sum_{i = 1}^{n}{{(X_{i} - \overline{X})}^{2},} X=n1∑i=1nXi,S2=n−11∑i=1n(Xi−X)2,则：

X ‾ ∼ N ( μ , σ 2 n ) \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right)\text{\ \ } X∼N(μ,nσ2) 或者 X ‾ − μ σ n ∼ N ( 0 , 1 ) \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1) n σX−μ∼N(0,1)

( n − 1 ) S 2 σ 2 = 1 σ 2 ∑ i = 1 n ( X i − X ‾ ) 2 ∼ χ 2 ( n − 1 ) \frac{(n - 1)S^{2}}{\sigma^{2}} = \frac{1}{\sigma^{2}}\sum_{i = 1}^{n}{{(X_{i} - \overline{X})}^{2}\sim\chi^{2}(n - 1)} σ2(n−1)S2=σ21∑i=1n(Xi−X)2∼χ2(n−1)

1 σ 2 ∑ i = 1 n ( X i − μ ) 2 ∼ χ 2 ( n ) \frac{1}{\sigma^{2}}\sum_{i = 1}^{n}{{(X_{i} - \mu)}^{2}\sim\chi^{2}(n)} σ21∑i=1n(Xi−μ)2∼χ2(n)

4) X ‾ − μ S / n ∼ t ( n − 1 ) \text{\ \ }\frac{\overline{X} - \mu}{S/\sqrt{n}}\sim t(n - 1) S/n X−μ∼t(n−1)

**4.重要公式与结论 **

(1)
对于 χ 2 ∼ χ 2 ( n ) \chi^{2}\sim\chi^{2}(n) χ2∼χ2(n)，有 E ( χ 2 ( n ) ) = n , D ( χ 2 ( n ) ) = 2 n ; E(\chi^{2}(n)) = n,D(\chi^{2}(n)) = 2n; E(χ2(n))=n,D(χ2(n))=2n;

(2) 对于 T ∼ t ( n ) T\sim t(n) T∼t(n)，有 E ( T ) = 0 , D ( T ) = n n − 2 ( n > 2 ) E(T) = 0,D(T) = \frac{n}{n - 2}(n > 2) E(T)=0,D(T)=n−2n(n>2)；

(3) 对于 F ~ F ( m , n ) F\tilde{\ }F(m,n) F ~F(m,n)，有
1 F ∼ F ( n , m ) , F a / 2 ( m , n ) = 1 F 1 − a / 2 ( n , m ) ; \frac{1}{F}\sim F(n,m),F_{a/2}(m,n) = \frac{1}{F_{1 - a/2}(n,m)}; F1∼F(n,m),Fa/2(m,n)=F1−a/2(n,m)1;

(4) 对于任意总体 X X X，有
E ( X ‾ ) = E ( X ) , E ( S 2 ) = D ( X ) , D ( X ‾ ) = D ( X ) n E(\overline{X}) = E(X),E(S^{2}) = D(X),D(\overline{X}) = \frac{D(X)}{n} E(X)=E(X),E(S2)=D(X),D(X)=nD(X)