机器学习的数学基础
高等数学
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 i j x i y j f(x_{1},x_{2},\cdots,x_{n}) = \sum_{i = 1}^{n}{\sum_{j =1}^{n}{a_{ {ij}}x_{i}y_{j}}} f(x1,x2,⋯,xn)=∑i=1n∑j=1naijxiyj，其中 a i j = a j i ( i , j = 1 , 2 , ⋯ , n ) a_{ {ij}} = a_{ {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 n n ] x = \ \begin{bmatrix}x_{1} \\ x_{1} \\ \vdots \\ x_{n} \\ \end{bmatrix},A = \begin{bmatrix} a_{11}& a_{12}& \cdots & a_{1n} \\ a_{21}& a_{22}& \cdots & a_{2n} \\ \cdots &\cdots &\cdots &\cdots \\ a_{n1}& a_{n2} & \cdots & a_{ {nn}} \\\end{bmatrix} x= ⎣ ⎡x1x1⋮xn⎦ ⎤,A=⎣ ⎡a11a21⋯an1a12a22⋯an2⋯⋯⋯⋯a1na2n⋯ann⎦ ⎤,这二次型 f f f可改写成矩阵向量形式 f = x T A x f =x^{T}{Ax} f=xTAx。其中 A A A称为二次型矩阵，因为 a i j = a j i ( i , j = 1 , 2 , ⋯ , n ) a_{ {ij}} =a_{ {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 A x f = \left( x_{1},x_{2},\cdots,x_{n} \right) =x^{T}{Ax} f=(x1,x2,⋯,xn)=xTAx经过合同变换 x = C y x = {Cy} x=Cy化为 f = x T A x = y T C T A C f = x^{T}{Ax} =y^{T}C^{T}{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 ⇒ k A ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow {kA}(k > 0),A^{T},A^{- 1},A^{*} ⇒kA(k>0),AT,A−1,A∗正定； ∣ A ∣ > 0 |A| >0 ∣A∣>0, A A A可逆； a i i > 0 a_{ {ii}} > 0 aii>0，且 ∣ A i i ∣ > 0 |A_{ {ii}}| > 0 ∣Aii∣>0
A A A， B B B正定 ⇒ A + B \Rightarrow A +B ⇒A+B正定，但 A B {AB} AB， B A {BA} BA不一定正定
A A A正定 ⇔ f ( x ) = x T A x > 0 , ∀ x ≠ 0 \Leftrightarrow f(x) = x^{T}{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 A Q = Q − 1 A Q = ( λ 1 ⋱ λ n ) , Q^{T}{AQ} = Q^{- 1}{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 ⇒ k A ( k > 0 ) , A T , A − 1 , A ∗ \Rightarrow {kA}(k >0),A^{T},A^{- 1},A^{*} ⇒kA(k>0),AT,A−1,A∗正定； ∣ A ∣ > 0 , A |A| > 0,A ∣A∣>0,A可逆； a i i > 0 a_{ {ii}} >0 aii>0，且 ∣ A i i ∣ > 0 |A_{ {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 A B {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 ˉ , B = A ˉ A\bigcap B=\varnothing ,A\bigcup B=\Omega ,A=\bar{B},B=\bar{A} A⋂B=∅,A⋃B=Ω,A=Bˉ,B=Aˉ
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)
(3) 分配律： ( A ⋂ B ) ⋂ C = A ⋂ ( B ⋂ C ) (A\bigcap B)\bigcap C=A\bigcap (B\bigcap C) (A⋂B)⋂C=A⋂(B⋂C)
3.德$\centerdot $摩根律
A ⋃ B ‾ = A ˉ ⋂ B ˉ \overline{A\bigcup B}=\bar{A}\bigcap \bar{B} A⋃B=Aˉ⋂Bˉ A ⋂ B ‾ = A ˉ ⋃ B ˉ \overline{A\bigcap B}=\bar{A}\bigcup \bar{B} A⋂B=Aˉ⋃Bˉ
4.完全事件组
A 1 A 2 ⋯ A n { {A}_{1}}{ {A}_{2}}\cdots { {A}_{n}} A1A2⋯An两两互斥，且和事件为必然事件，即${ {A}{i}}\bigcap { {A}{j}}=\varnothing, i\ne j ,\underset{i=1}{\overset{n}{\mathop \bigcup }},=\Omega $
5.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)=\sum\limits_{i=1}^{n}{P(A|{
{B}{i}})P({
{B}{i}}),{
{B}{i}}{
{B}{j}}}=\varnothing ,i\ne j,\underset{i=1}{\overset{n}{\mathop{\bigcup }}},{
{B}_{i}}=\Omega $
(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\limits_{i=1}^{n}{P(A|{
{B}_{i}})P({
{B}_{i}})}},j=1,2,\cdots ,n P(Bj∣A)=i=1∑nP(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)
6.事件的独立性
(1) A A A与 B B B相互独立 ⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) ⇔P(AB)=P(A)P(B)
(2) A A A， B B B， C C C两两独立
⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) ⇔P(AB)=P(A)P(B); P ( B C ) = P ( B ) P ( C ) P(BC)=P(B)P(C) P(BC)=P(B)P(C) ; P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P(C) P(AC)=P(A)P(C);
(3) A A A， B B B， C C C相互独立
⇔ P ( A B ) = P ( A ) P ( B ) \Leftrightarrow P(AB)=P(A)P(B) ⇔P(AB)=P(A)P(B); P ( B C ) = P ( B ) P ( C ) P(BC)=P(B)P(C) P(BC)=P(B)P(C) ;
P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P(C) P(AC)=P(A)P(C) ; P ( A B C ) = P ( A ) P ( B ) P ( C ) P(ABC)=P(A)P(B)P(C) P(ABC)=P(A)P(B)P(C)
7.独立重复试验
将某试验独立重复 n n n次，若每次实验中事件A发生的概率为 p p p，则 n n n次试验中 A A A发生 k k k次的概率为：
P ( X = k ) = C n k p k ( 1 − p ) n − k P(X=k)=C_{n}^{k}{
{p}^{k}}{
{(1-p)}^{n-k}} P(X=k)=Cnkpk(1−p)n−k
8.重要公式与结论
( 1 ) P ( A ˉ ) = 1 − P ( A ) (1)P(\bar{A})=1-P(A) (1)P(Aˉ)=1−P(A)
( 2 ) P ( A ⋃ B ) = P ( A ) + P ( B ) − P ( A B ) (2)P(A\bigcup B)=P(A)+P(B)-P(AB) (2)P(A⋃B)=P(A)+P(B)−P(AB)
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)
( 3 ) P ( A − B ) = P ( A ) − P ( A B ) (3)P(A-B)=P(A)-P(AB) (3)P(A−B)=P(A)−P(AB)
( 4 ) P ( A B ˉ ) = P ( A ) − P ( A B ) , P ( A ) = P ( A B ) + P ( A B ˉ ) , (4)P(A\bar{B})=P(A)-P(AB),P(A)=P(AB)+P(A\bar{B}), (4)P(ABˉ)=P(A)−P(AB),P(A)=P(AB)+P(ABˉ),
P ( A ⋃ B ) = P ( A ) + P ( A ˉ B ) = P ( A B ) + P ( A B ˉ ) + P ( A ˉ B ) P(A\bigcup B)=P(A)+P(\bar{A}B)=P(AB)+P(A\bar{B})+P(\bar{A}B) P(A⋃B)=P(A)+P(AˉB)=P(AB)+P(ABˉ)+P(AˉB)
(5)条件概率 P ( ⋅ ∣ B ) P(\centerdot |B) P(⋅∣B)满足概率的所有性质，
例如：. P ( A ˉ 1 ∣ B ) = 1 − P ( A 1 ∣ B ) P({
{\bar{A}}_{1}}|B)=1-P({
{A}_{1}}|B) P(Aˉ1∣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\limits_{i=1}^{n}{
{
{A}_{i}}})=\prod\limits_{i=1}^{n}{P({
{A}_{i}})}, P(i=1⋂nAi)=i=1∏nP(Ai),
P ( ⋃ i = 1 n A i ) = ∏ i = 1 n ( 1 − P ( A i ) ) P(\bigcup\limits_{i=1}^{n}{
{
{A}_{i}}})=\prod\limits_{i=1}^{n}{(1-P({
{A}_{i}}))} P(i=1⋃nAi)=i=1∏n(1−P(Ai))
(7)互斥、互逆与独立性之间的关系：
A A A与 B B B互逆 ⇒ \Rightarrow ⇒ A A A与 B B B互斥，但反之不成立， A A A与 B B B互斥（或互逆）且均非零概率事件$\Rightarrow $ A A A与 B B 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(\centerdot ),g(\centerdot ) 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 ) d x = 1 \int_{- \infty}^{+\infty}{f(x){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 ) d t F(x) = \int_{- \infty}^{x}{f(t){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) = { \begin{matrix} & \frac{1}{b - a},a < x< b \ & 0, \ \end{matrix} $
(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)=2πσ1e−2σ2(x−μ)2,σ>0,∞<x<+∞
(6)指数分布:$E(\lambda):f(x) ={ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \ & 0, \ \end{matrix} $
(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)=2π1,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 i j ; i , j = 1 , 2 , ⋯ P\{ X = x_{i},Y = y_{j}\} = p_{ {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 i j , i = 1 , 2 , ⋯ p_{i \cdot} = \sum_{j = 1}^{\infty}p_{ {ij}},i =1,2,\cdots pi⋅=∑j=1∞pij,i=1,2,⋯ p ⋅ j = ∑ i ∞ p i j , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{ {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 i j p ⋅ j P\{ X = x_{i}|Y = y_{j}\} = \frac{p_{
{ij}}}{p_{\cdot j}} P{
X=xi∣Y=yj}=p⋅jpij
P { Y = y j ∣ X = x i } = p i j p i ⋅ P\{ Y = y_{j}|X = x_{i}\} = \frac{p_{
{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):
-
f ( x , y ) ≥ 0 f(x,y) \geq 0 f(x,y)≥0
-
∫ − ∞ + ∞ ∫ − ∞ + ∞ 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 ) d y f_{X}\left( x \right) = \int_{- \infty}^{+ \infty}{f\left( x,y \right){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 ) = { 1 S ( D ) , ( x , y ) ∈ D 0 , 其他 f(x,y) = \begin{cases} \frac{1}{S(D)},(x,y) \in D \\ 0,其他 \end{cases} f(x,y)={ S(D)1,(x,y)∈D0,其他
(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), ( 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−ρ21.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 i j = p i ⋅ ⋅ p ⋅ j \Leftrightarrow p_{
{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 X Y = 0 \rho_{
{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 i j , Z = g ( X , Y ) P\left( X = x_{i},Y = y_{i} \right) = p_{ {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 ) d x d y P\left\{ \left( X,Y \right) \in D \right\} = \iint_{D}^{}{f\left( x,y \right){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).
-
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)
-
X {\ X} X关于 Y = y Y=y 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))
-
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,C12σ12C22σ22).
(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 X X和 Y Y 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 i j \left( X,Y \right)\sim P\{ X = x_{i},Y = y_{j}\} = p_{ {ij}} (X,Y)∼P{ X=xi,Y=yj}=pij; E ( Z ) = ∑ i ∑ j g ( x i , y j ) p i j E(Z) = \sum_{i}^{}{\sum_{j}^{}{g(x_{i},y_{j})p_{ {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 X Y = C o v ( X , Y ) D ( X ) D ( Y ) \rho_{
{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/nX∼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 F 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：
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X ‾ ∼ N ( μ , σ 2 n ) \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ } 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) nσX−μ∼N(0,1)
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( 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)
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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 ) {\ \ }\frac{\overline{X} - \mu}{S/\sqrt{n}}\sim t(n - 1) S/nX−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)