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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 BA⊂B，若A AA发生，则B BB发生。 Notes：A AA是B BB的子事件，P ( A ) ≤ P ( B ) P(A) \le P(B)P(A)≤P(B)

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

(3) 和事件：A ⋃ B A\bigcup BA⋃B（或A + B A + BA+B），A AA与B BB中至少有一个发生。 推广：若A 1 , A 2 , … … A_{1},A_{2},……A

1

​

,A

2

​

,……是互不相容的事件序列，则P ( A 1 ⋂ A 2 ) = P ( A 1 ) + P ( A 2 ) + … … P(A_{1}\bigcap A_{2})=P(A_{1})+P(A_{2})+……P(A

1

​

⋂A

2

​

)=P(A

1

​

)+P(A

2

​

)+……

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

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

(6) 互斥事件（互不相容）：A ⋂ B A\bigcap BA⋂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=

B

ˉ

,B=

A

ˉ

(8)P ( Ω ) = 1 P(\Omega)=1P(Ω)=1

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 AA⋃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

​

A

ˉ

⋂

B

ˉ

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

A⋂B

​

A

ˉ

⋃

B

ˉ

4.完全事件组

A 1 A 2 ⋯ A n {{A}{1}}{{A}{2}}\cdots {{A}{n}}A

1

​

A

2

​

⋯A

n

​

两两互斥，且和事件为必然事件，即A i ∩ A j = ∅ , i ≠ j , ⋃ i = 1 n = Ω . {A{i}} \cap A_{j}=\varnothing, i \neq j, \bigcup_{i=1}^{n}=\Omega\ .A

i

​

∩A

j

​

=∅,i



​

=j,⋃

i=1

n

​

=Ω .

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(A\bigcap B)}{P(A)}P(B∣A)=

P(A)

P(A⋂B)

​

,表示A AA发生的条件下，B BB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\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}}=\OmegaP(A)=

i=1

∑

n

​

P(A∣B

i

​

)P(B

i

​

),B

i

​

B

j

​

=∅,i



​

=j,

i=1

⋃

n

​

​

B

i

​

=Ω 如何推出？ 条件概率变形，P ( A ⋂ B ) = P ( B ∣ A ) P ( A ) P(A\bigcap B)=P(B|A)P(A)P(A⋂B)=P(B∣A)P(A)

(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 ,nP(B

j

​

∣A)=

i=1

∑

n

​

P(A∣B

i

​

)P(B

i

​

)

P(A∣B

j

​

)P(B

j

​

)

​

,j=1,2,⋯,n

注：上述公式中事件B i {{B}{i}}B

i

​

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(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 ⋯ 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(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

​

)

6.事件的独立性

(1)A AA与B BB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 AA，B BB，C CC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©P(BC)=P(B)P© ;P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P©P(AC)=P(A)P©;

(3)A AA，B BB，C CC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©P(BC)=P(B)P© ;

P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P©P(AC)=P(A)P© ; P ( A B C ) = P ( A ) P ( B ) P ( C ) P(ABC)=P(A)P(B)P©P(ABC)=P(A)P(B)P©

7.独立重复试验

将某试验独立重复n nn次，若每次实验中事件A发生的概率为p pp，则n nn次试验中A AA发生k kk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)=C

n

k

​

p

k

(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(

A

ˉ

)=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©-P(AB)-P(BC)-P(AC)+P(ABC)P(A⋃B⋃C)=P(A)+P(B)+P©−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(A

B

ˉ

)=P(A)−P(AB),P(A)=P(AB)+P(A

B

ˉ

),

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(

A

ˉ

B)=P(AB)+P(A

B

ˉ

)+P(

A

ˉ

B)

(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(

A

ˉ

1

​

∣B)=1−P(A

1

​

∣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(A

1

​

⋃A

2

​

∣B)=P(A

1

​

∣B)+P(A

2

​

∣B)−P(A

1

​

A

2

​

∣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(A

1

​

A

2

​

∣B)=P(A

1

​

∣B)P(A

2

​

∣A

1

​

B)

(6)若A 1 , A 2 , ⋯   , A n {{A}{1}},{{A}{2}},\cdots ,{{A}{n}}A

1

​

,A

2

​

,⋯,A

n

​

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

⋂

n

​

A

i

​

)=

i=1

∏

n

​

P(A

i

​

),

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

⋃

n

​

A

i

​

)=

i=1

∏

n

​

(1−P(A

i

​

))

(7)互斥、互逆与独立性之间的关系：

A AA与B BB互逆⇒ \Rightarrow⇒ A AA与B BB互斥，但反之不成立，A AA与B BB互斥（或互逆）且均非零概率事件$\Rightarrow $A AA与B BB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}}A

1

​

,A

2

​

,⋯,A

m

​

,B

1

​

,B

2

​

,⋯,B

n

​

相互独立，则f ( A 1 , A 2 , ⋯   , A m ) f({{A}{1}},{{A}{2}},\cdots ,{{A}{m}})f(A

1

​

,A

2

​

,⋯,A

m

​

)与g ( B 1 , B 2 , ⋯   , B n ) g({{B}{1}},{{B}{2}},\cdots ,{{B}_{n}})g(B

1

​

,B

2

​

,⋯,B

n

​

)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 < + \inftyF(x)=P(X≤x),−∞<x<+∞

性质：(1)0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 10≤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) = 1F(−∞)=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} = 1P(X=x

i

​

)=p

i

​

,i=1,2,⋯,n,⋯p

i

​

≥0,∑

i=1

∞

​

p

i

​

=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 xx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)=∫

−∞

x

​

f(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 = 0,1P(X=k)=p

k

(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,nP(X=k)=C

n

k

​

p

k

(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\cdotsP(X=k)=

k!

λ

k

​

e

−λ

,λ>0,k=0,1,2⋯

(4) 均匀分布U ( a , b ) U(a,b)U(a,b)：f ( x ) = { 1 b − a , a < x < b 0 , f(x) = {

1b−a,a<x<b0,

1b−a,a<x<b0,

f(x)={

​

b−a

1

​

,a<x<b

0,

​

(5) 正态分布:N ( μ , σ 2 ) : N(\mu,\sigma^{2}):N(μ,σ

2

): 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 < + \inftyφ(x)=

2π

​

σ

1

​

e

−

2σ

2

(x−μ)

2

​

,σ>0,∞<x<+∞

(6)指数分布:E ( λ ) : f ( x ) = { λ e − λ x , x > 0 , λ > 0 0 , E(\lambda):f(x) ={

λe−λx,x>0,λ>00,

λe−λx,x>0,λ>00,

E(λ):f(x)={

​

λe

−λx

,x>0,λ>0

0,

​

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

k−1

p,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)😛(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)😛(X=k)=

C

N

n

​

C

M

k

​

C

N−M

n−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=x

1

​

)=p

i

​

,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=y

j

​

)=∑

g(x

i

​

)=y

i

​

​

P(X=x

i

​

)

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

~

f

X

​

(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}F

y

​

(y)=P(Y≤y)=P(g(X)≤y)=∫

g(x)≤y

​

f

x

​

(x)dx， f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F’_{Y}(y)f

Y

​

(y)=F

Y

′

​

(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)=

2

1

​

, 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,σ

2

)⇒

σ

X−μ

​

∼N(0,1),P(X≤a)=Φ(

σ

a−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§ \Rightarrow P(X = m + k|X > m) = P(X = k)X∼G§⇒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,\cdotsP{X=x

i

​

,Y=y

j

​

}=p

ij

​

;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,\cdotsp

i⋅

​

=∑

j=1

∞

​

p

ij

​

,i=1,2,⋯ p ⋅ j = ∑ i ∞ p i j , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{{ij}},j = 1,2,\cdotsp

⋅j

​

=∑

i

∞

​

p

ij

​

,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=x

i

​

∣Y=y

j

​

}=

p

⋅j

​

p

ij

​

​

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=y

j

​

∣X=x

i

​

}=

p

i⋅

​

p

ij

​

​

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 0f(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

​

∫

−∞

y

​

f(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}}f

X

​

(x)=∫

−∞

+∞

​

f(x,y)dy f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}f

Y

​

(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)}f

X∣Y

​

(x∣y)=

f

Y

​

(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)}f

Y∣X

​

(y∣x)=

f

X

​

(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) =

{1S(D),(x,y)∈D0,其他

{1S(D),(x,y)∈D0,其他

f(x,y)={

S(D)

1

​

,(x,y)∈D

0,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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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)=

2πσ

1

​

σ

2

​

1−ρ

2

​

1

​

.exp{

2(1−ρ

2

)

−1

​

[

σ

1

2

​

(x−μ

1

​

)

2

​

−2ρ

σ

1

​

σ

2

​

(x−μ

1

​

)(y−μ

2

​

)

​

+

σ

2

2

​

(y−μ

2

​

)

2

​

]}

5.随机变量的独立性和相关性

X XX和Y YY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)=F

X

​

(x)F

Y

​

(y):

⇔ p i j = p i ⋅ ⋅ p ⋅ j \Leftrightarrow p_{{ij}} = p_{i \cdot} \cdot p_{\cdot j}⇔p

ij

​

=p

i⋅

​

⋅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)=f

X

​

(x)f

Y

​

(y)（连续型）

X XX和Y YY的相关性：

相关系数ρ X Y = 0 \rho_{{XY}} = 0ρ

XY

​

=0时，称X XX和Y YY不相关，

否则称X XX和Y YY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=x

i

​

,Y=y

i

​

)=p

ij

​

,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=z

k

​

)=P{g(X,Y)=z

k

​

}=∑

g(x

i

​

,y

i

​

)=z

k

​

​

P(X=x

i

​

,Y=y

j

​

)

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}F

z

​

(z)=P{g(X,Y)≤z}=∬

g(x,y)≤z

​

f(x,y)dxdy，f z ( z ) = F z ′ ( z ) f_{z}(z) = F’_{z}(z)f

z

​

(z)=F

z

′

​

(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,}f

X

​

(x)=∫

−∞

+∞

​

f(x,y)dy,

f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}f

Y

​

(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}=∬

D

​

f(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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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(μ

1

​

,σ

1

2

​

),Y∼N(μ

2

​

,σ

2

2

​

).

X XX与Y YY相互独立⇔ ρ = 0 \Leftrightarrow \rho = 0⇔ρ=0，即X XX与Y YY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)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

​

+2C

1

​

C

2

​

σ

1

​

σ

2

​

ρ)

X {\ X} X关于Y = y Y=yY=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(μ

1

​

+ρ

σ

2

​

σ

1

​

​

(y−μ

2

​

),σ

1

2

​

(1−ρ

2

))

Y YY关于X = x X = xX=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(μ

2

​

+ρ

σ

1

​

σ

2

​

​

(x−μ

1

​

),σ

2

2

​

(1−ρ

2

))

(4) 若X XX与Y YY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(μ

1

​

,σ

1

2

​

),N(μ

1

​

,σ

2

2

​

),

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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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}).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

​

).

(5) 若X XX与Y YY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。

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 XX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=x

i

​

}=p

i

​

,E(Y)=∑

i

​

g(x

i

​

)p

i

​

；

X XX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=x

i

​

,Y=y

j

​

}=p

ij

​

; 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

​

∑

j

​

g(x

i

​

,y

j

​

)p

ij

​

( 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\rbrackCov(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)}}ρ

XY

​

D(X)

​

D(Y)

​

Cov(X,Y)

​

,k kk阶原点矩 E ( X k ) E(X^{k})E(X

k

);

k kk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(X

1

​

+X

2

​

,Y)=Cov(X

1

​

,Y)+Cov(X

2

​

,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 > 0a>0

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

，其中a < 0 a < 0a<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(X

2

)−E

2

(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) = 1ρ(X,Y)=1⇔P(Y=aX+b)=1，其中a > 0 a > 0a>0

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

(4) 下面5个条件互为充要条件：

ρ ( X , Y ) = 0 \rho(X,Y) = 0ρ(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 XX与Y YY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 XX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 XX的n nn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}X

1

​

,X

2

​

⋯,X

n

​

，称为容量为n nn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},X

1

​

,X

2

​

⋯,X

n

​

,是来自总体X XX的一个样本，g ( X 1 , X 2 ⋯   , X n ) g(X_{1},X_{2}\cdots,X_{n})g(X

1

​

,X

2

​

⋯,X

n

​

)）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(X

1

​

,X

2

​

⋯,X

n

​

)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

n

1

​

∑

i=1

n

​

X

i

​

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}S

2

n−1

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

样本矩：样本k kk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,\cdotsA

k

​

n

1

​

∑

i=1

n

​

X

i

k

​

,k=1,2,⋯

样本k kk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,\cdotsB

k

​

n

1

​

∑

i=1

n

​

(X

i

​

−

X

)

k

,k=1,2,⋯

2.分布

χ 2 \chi^{2}χ

2

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)χ

2

=X

1

2

​

+X

2

2

​

+⋯+X

n

2

​

∼χ

2

(n)，其中X 1 , X 2 ⋯   , X n , X_{1},X_{2}\cdots,X_{n},X

1

​

,X

2

​

⋯,X

n

​

,相互独立，且同服从N ( 0 , 1 ) N(0,1)N(0,1)

t tt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∼χ

2

(n),且X XX，Y YY 相互独立。

F FF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/n

2

​

X/n

1

​

​

∼F(n

1

​

,n

2

​

)，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∼χ

2

(n

1

​

),Y∼χ

2

(n

2

​

),且X XX，Y YY相互独立。

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

α

​

)=α,则称x α x_{\alpha}x

α

​

为X XX的α \alphaα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}X

1

​

,X

2

​

⋯,X

n

​

为来自正态总体N ( μ , σ 2 ) N(\mu,\sigma^{2})N(μ,σ

2

)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

n

1

​

∑

i=1

n

​

X

i

​

,S

2

n−1

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

,则：

X ‾ ∼ N ( μ , σ 2 n ) \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ }

X

∼N(μ,

n

σ

2

​

) 或者X ‾ − μ σ n ∼ N ( 0 , 1 ) \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)

n

​

σ

​

X

−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)}

σ

2

(n−1)S

2

​

σ

2

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

∼χ

2

(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)}

σ

2

1

​

∑

i=1

n

​

(X

i

​

−μ)

2

∼χ

2

(n)

4) X ‾ − μ S / n ∼ t ( n − 1 ) {\ \ }\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)χ

2

∼χ

2

(n)，有E ( χ 2 ( n ) ) = n , D ( χ 2 ( n ) ) = 2 n ; E(\chi^{2}(n)) = n,D(\chi^{2}(n)) = 2n;E(χ

2

(n))=n,D(χ

2

(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−2

n

​

(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)};

F

1

​

∼F(n,m),F

a/2

​

(m,n)=

F

1−a/2

​

(n,m)

1

​

;

(4) 对于任意总体X XX，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(S

2

)=D(X),D(

X

)=

n

D(X)

​

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 BA⊂B，若A AA发生，则B BB发生。 Notes：A AA是B BB的子事件，P ( A ) ≤ P ( B ) P(A) \le P(B)P(A)≤P(B)

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

(3) 和事件：A ⋃ B A\bigcup BA⋃B（或A + B A + BA+B），A AA与B BB中至少有一个发生。 推广：若A 1 , A 2 , … … A_{1},A_{2},……A

1

​

,A

2

​

,……是互不相容的事件序列，则P ( A 1 ⋂ A 2 ) = P ( A 1 ) + P ( A 2 ) + … … P(A_{1}\bigcap A_{2})=P(A_{1})+P(A_{2})+……P(A

1

​

⋂A

2

​

)=P(A

1

​

)+P(A

2

​

)+……

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

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

(6) 互斥事件（互不相容）：A ⋂ B A\bigcap BA⋂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=

B

ˉ

,B=

A

ˉ

(8)P ( Ω ) = 1 P(\Omega)=1P(Ω)=1

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 AA⋃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

​

A

ˉ

⋂

B

ˉ

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

A⋂B

​

A

ˉ

⋃

B

ˉ

4.完全事件组

A 1 A 2 ⋯ A n {{A}{1}}{{A}{2}}\cdots {{A}{n}}A

1

​

A

2

​

⋯A

n

​

两两互斥，且和事件为必然事件，即A i ∩ A j = ∅ , i ≠ j , ⋃ i = 1 n = Ω . {A{i}} \cap A_{j}=\varnothing, i \neq j, \bigcup_{i=1}^{n}=\Omega\ .A

i

​

∩A

j

​

=∅,i



​

=j,⋃

i=1

n

​

=Ω .

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(A\bigcap B)}{P(A)}P(B∣A)=

P(A)

P(A⋂B)

​

,表示A AA发生的条件下，B BB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\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}}=\OmegaP(A)=

i=1

∑

n

​

P(A∣B

i

​

)P(B

i

​

),B

i

​

B

j

​

=∅,i



​

=j,

i=1

⋃

n

​

​

B

i

​

=Ω 如何推出？ 条件概率变形，P ( A ⋂ B ) = P ( B ∣ A ) P ( A ) P(A\bigcap B)=P(B|A)P(A)P(A⋂B)=P(B∣A)P(A)

(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 ,nP(B

j

​

∣A)=

i=1

∑

n

​

P(A∣B

i

​

)P(B

i

​

)

P(A∣B

j

​

)P(B

j

​

)

​

,j=1,2,⋯,n

注：上述公式中事件B i {{B}{i}}B

i

​

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(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 ⋯ 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(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

​

)

6.事件的独立性

(1)A AA与B BB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 AA，B BB，C CC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©P(BC)=P(B)P© ;P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P©P(AC)=P(A)P©;

(3)A AA，B BB，C CC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©P(BC)=P(B)P© ;

P ( A C ) = P ( A ) P ( C ) P(AC)=P(A)P©P(AC)=P(A)P© ; P ( A B C ) = P ( A ) P ( B ) P ( C ) P(ABC)=P(A)P(B)P©P(ABC)=P(A)P(B)P©

7.独立重复试验

将某试验独立重复n nn次，若每次实验中事件A发生的概率为p pp，则n nn次试验中A AA发生k kk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)=C

n

k

​

p

k

(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(

A

ˉ

)=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©-P(AB)-P(BC)-P(AC)+P(ABC)P(A⋃B⋃C)=P(A)+P(B)+P©−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(A

B

ˉ

)=P(A)−P(AB),P(A)=P(AB)+P(A

B

ˉ

),

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(

A

ˉ

B)=P(AB)+P(A

B

ˉ

)+P(

A

ˉ

B)

(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(

A

ˉ

1

​

∣B)=1−P(A

1

​

∣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(A

1

​

⋃A

2

​

∣B)=P(A

1

​

∣B)+P(A

2

​

∣B)−P(A

1

​

A

2

​

∣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(A

1

​

A

2

​

∣B)=P(A

1

​

∣B)P(A

2

​

∣A

1

​

B)

(6)若A 1 , A 2 , ⋯   , A n {{A}{1}},{{A}{2}},\cdots ,{{A}{n}}A

1

​

,A

2

​

,⋯,A

n

​

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

⋂

n

​

A

i

​

)=

i=1

∏

n

​

P(A

i

​

),

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

⋃

n

​

A

i

​

)=

i=1

∏

n

​

(1−P(A

i

​

))

(7)互斥、互逆与独立性之间的关系：

A AA与B BB互逆⇒ \Rightarrow⇒ A AA与B BB互斥，但反之不成立，A AA与B BB互斥（或互逆）且均非零概率事件$\Rightarrow $A AA与B BB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}}A

1

​

,A

2

​

,⋯,A

m

​

,B

1

​

,B

2

​

,⋯,B

n

​

相互独立，则f ( A 1 , A 2 , ⋯   , A m ) f({{A}{1}},{{A}{2}},\cdots ,{{A}{m}})f(A

1

​

,A

2

​

,⋯,A

m

​

)与g ( B 1 , B 2 , ⋯   , B n ) g({{B}{1}},{{B}{2}},\cdots ,{{B}_{n}})g(B

1

​

,B

2

​

,⋯,B

n

​

)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 < + \inftyF(x)=P(X≤x),−∞<x<+∞

性质：(1)0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 10≤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) = 1F(−∞)=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} = 1P(X=x

i

​

)=p

i

​

,i=1,2,⋯,n,⋯p

i

​

≥0,∑

i=1

∞

​

p

i

​

=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 xx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)=∫

−∞

x

​

f(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 = 0,1P(X=k)=p

k

(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,nP(X=k)=C

n

k

​

p

k

(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\cdotsP(X=k)=

k!

λ

k

​

e

−λ

,λ>0,k=0,1,2⋯

(4) 均匀分布U ( a , b ) U(a,b)U(a,b)：f ( x ) = { 1 b − a , a < x < b 0 , f(x) = {

1b−a,a<x<b0,

1b−a,a<x<b0,

f(x)={

​

b−a

1

​

,a<x<b

0,

​

(5) 正态分布:N ( μ , σ 2 ) : N(\mu,\sigma^{2}):N(μ,σ

2

): 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 < + \inftyφ(x)=

2π

​

σ

1

​

e

−

2σ

2

(x−μ)

2

​

,σ>0,∞<x<+∞

(6)指数分布:E ( λ ) : f ( x ) = { λ e − λ x , x > 0 , λ > 0 0 , E(\lambda):f(x) ={

λe−λx,x>0,λ>00,

λe−λx,x>0,λ>00,

E(λ):f(x)={

​

λe

−λx

,x>0,λ>0

0,

​

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

k−1

p,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)😛(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)😛(X=k)=

C

N

n

​

C

M

k

​

C

N−M

n−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=x

1

​

)=p

i

​

,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=y

j

​

)=∑

g(x

i

​

)=y

i

​

​

P(X=x

i

​

)

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

~

f

X

​

(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}F

y

​

(y)=P(Y≤y)=P(g(X)≤y)=∫

g(x)≤y

​

f

x

​

(x)dx， f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F’_{Y}(y)f

Y

​

(y)=F

Y

′

​

(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)=

2

1

​

, 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,σ

2

)⇒

σ

X−μ

​

∼N(0,1),P(X≤a)=Φ(

σ

a−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§ \Rightarrow P(X = m + k|X > m) = P(X = k)X∼G§⇒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,\cdotsP{X=x

i

​

,Y=y

j

​

}=p

ij

​

;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,\cdotsp

i⋅

​

=∑

j=1

∞

​

p

ij

​

,i=1,2,⋯ p ⋅ j = ∑ i ∞ p i j , j = 1 , 2 , ⋯ p_{\cdot j} = \sum_{i}^{\infty}p_{{ij}},j = 1,2,\cdotsp

⋅j

​

=∑

i

∞

​

p

ij

​

,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=x

i

​

∣Y=y

j

​

}=

p

⋅j

​

p

ij

​

​

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=y

j

​

∣X=x

i

​

}=

p

i⋅

​

p

ij

​

​

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 0f(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

​

∫

−∞

y

​

f(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}}f

X

​

(x)=∫

−∞

+∞

​

f(x,y)dy f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}f

Y

​

(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)}f

X∣Y

​

(x∣y)=

f

Y

​

(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)}f

Y∣X

​

(y∣x)=

f

X

​

(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) =

{1S(D),(x,y)∈D0,其他

{1S(D),(x,y)∈D0,其他

f(x,y)={

S(D)

1

​

,(x,y)∈D

0,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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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)=

2πσ

1

​

σ

2

​

1−ρ

2

​

1

​

.exp{

2(1−ρ

2

)

−1

​

[

σ

1

2

​

(x−μ

1

​

)

2

​

−2ρ

σ

1

​

σ

2

​

(x−μ

1

​

)(y−μ

2

​

)

​

+

σ

2

2

​

(y−μ

2

​

)

2

​

]}

5.随机变量的独立性和相关性

X XX和Y YY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)=F

X

​

(x)F

Y

​

(y):

⇔ p i j = p i ⋅ ⋅ p ⋅ j \Leftrightarrow p_{{ij}} = p_{i \cdot} \cdot p_{\cdot j}⇔p

ij

​

=p

i⋅

​

⋅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)=f

X

​

(x)f

Y

​

(y)（连续型）

X XX和Y YY的相关性：

相关系数ρ X Y = 0 \rho_{{XY}} = 0ρ

XY

​

=0时，称X XX和Y YY不相关，

否则称X XX和Y YY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=x

i

​

,Y=y

i

​

)=p

ij

​

,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=z

k

​

)=P{g(X,Y)=z

k

​

}=∑

g(x

i

​

,y

i

​

)=z

k

​

​

P(X=x

i

​

,Y=y

j

​

)

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}F

z

​

(z)=P{g(X,Y)≤z}=∬

g(x,y)≤z

​

f(x,y)dxdy，f z ( z ) = F z ′ ( z ) f_{z}(z) = F’_{z}(z)f

z

​

(z)=F

z

′

​

(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,}f

X

​

(x)=∫

−∞

+∞

​

f(x,y)dy,

f Y ( y ) = ∫ − ∞ + ∞ f ( x , y ) d x f_{Y}(y) = \int_{- \infty}^{+ \infty}{f(x,y)dx}f

Y

​

(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}=∬

D

​

f(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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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(μ

1

​

,σ

1

2

​

),Y∼N(μ

2

​

,σ

2

2

​

).

X XX与Y YY相互独立⇔ ρ = 0 \Leftrightarrow \rho = 0⇔ρ=0，即X XX与Y YY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)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

​

+2C

1

​

C

2

​

σ

1

​

σ

2

​

ρ)

X {\ X} X关于Y = y Y=yY=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(μ

1

​

+ρ

σ

2

​

σ

1

​

​

(y−μ

2

​

),σ

1

2

​

(1−ρ

2

))

Y YY关于X = x X = xX=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(μ

2

​

+ρ

σ

1

​

σ

2

​

​

(x−μ

1

​

),σ

2

2

​

(1−ρ

2

))

(4) 若X XX与Y YY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(μ

1

​

,σ

1

2

​

),N(μ

1

​

,σ

2

2

​

),

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(μ

1

​

,μ

2

​

,σ

1

2

​

,σ

2

2

​

,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}).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

​

).

(5) 若X XX与Y YY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。

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 XX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=x

i

​

}=p

i

​

,E(Y)=∑

i

​

g(x

i

​

)p

i

​

；

X XX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=x

i

​

,Y=y

j

​

}=p

ij

​

; 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

​

∑

j

​

g(x

i

​

,y

j

​

)p

ij

​

( 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\rbrackCov(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)}}ρ

XY

​

D(X)

​

D(Y)

​

Cov(X,Y)

​

,k kk阶原点矩 E ( X k ) E(X^{k})E(X

k

);

k kk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(X

1

​

+X

2

​

,Y)=Cov(X

1

​

,Y)+Cov(X

2

​

,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 > 0a>0

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

，其中a < 0 a < 0a<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(X

2

)−E

2

(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) = 1ρ(X,Y)=1⇔P(Y=aX+b)=1，其中a > 0 a > 0a>0

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

(4) 下面5个条件互为充要条件：

ρ ( X , Y ) = 0 \rho(X,Y) = 0ρ(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 XX与Y YY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 XX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 XX的n nn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}X

1

​

,X

2

​

⋯,X

n

​

，称为容量为n nn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},X

1

​

,X

2

​

⋯,X

n

​

,是来自总体X XX的一个样本，g ( X 1 , X 2 ⋯   , X n ) g(X_{1},X_{2}\cdots,X_{n})g(X

1

​

,X

2

​

⋯,X

n

​

)）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(X

1

​

,X

2

​

⋯,X

n

​

)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

n

1

​

∑

i=1

n

​

X

i

​

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}S

2

n−1

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

样本矩：样本k kk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,\cdotsA

k

​

n

1

​

∑

i=1

n

​

X

i

k

​

,k=1,2,⋯

样本k kk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,\cdotsB

k

​

n

1

​

∑

i=1

n

​

(X

i

​

−

X

)

k

,k=1,2,⋯

2.分布

χ 2 \chi^{2}χ

2

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)χ

2

=X

1

2

​

+X

2

2

​

+⋯+X

n

2

​

∼χ

2

(n)，其中X 1 , X 2 ⋯   , X n , X_{1},X_{2}\cdots,X_{n},X

1

​

,X

2

​

⋯,X

n

​

,相互独立，且同服从N ( 0 , 1 ) N(0,1)N(0,1)

t tt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∼χ

2

(n),且X XX，Y YY 相互独立。

F FF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/n

2

​

X/n

1

​

​

∼F(n

1

​

,n

2

​

)，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∼χ

2

(n

1

​

),Y∼χ

2

(n

2

​

),且X XX，Y YY相互独立。

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

α

​

)=α,则称x α x_{\alpha}x

α

​

为X XX的α \alphaα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}X

1

​

,X

2

​

⋯,X

n

​

为来自正态总体N ( μ , σ 2 ) N(\mu,\sigma^{2})N(μ,σ

2

)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

n

1

​

∑

i=1

n

​

X

i

​

,S

2

n−1

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

,则：

X ‾ ∼ N ( μ , σ 2 n ) \overline{X}\sim N\left( \mu,\frac{\sigma^{2}}{n} \right){\ \ }

X

∼N(μ,

n

σ

2

​

) 或者X ‾ − μ σ n ∼ N ( 0 , 1 ) \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}\sim N(0,1)

n

​

σ

​

X

−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)}

σ

2

(n−1)S

2

​

σ

2

1

​

∑

i=1

n

​

(X

i

​

−

X

)

2

∼χ

2

(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)}

σ

2

1

​

∑

i=1

n

​

(X

i

​

−μ)

2

∼χ

2

(n)

4) X ‾ − μ S / n ∼ t ( n − 1 ) {\ \ }\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)χ

2

∼χ

2

(n)，有E ( χ 2 ( n ) ) = n , D ( χ 2 ( n ) ) = 2 n ; E(\chi^{2}(n)) = n,D(\chi^{2}(n)) = 2n;E(χ

2

(n))=n,D(χ

2

(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−2

n

​

(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)};

F

1

​

∼F(n,m),F

a/2

​

(m,n)=

F

1−a/2

​

(n,m)

1

​

;

(4) 对于任意总体X XX，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(S

2

)=D(X),D(

X

)=

n

D(X)

​