【概率论】3.3数学期望、方差与协方差
数学期望、方差与协方差
- 1.数学期望
- 1.定义
- 2.性质
- 2.方差
- 1.定义
- 2.计算公式
- 3.性质
- 3.常见分布的期望和方差
- 4.协方差
- 1.相关系数
- 2.协方差矩阵
- 5.重要公式与结论
1.数学期望
1.定义
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
2.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
1.定义
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
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
上面这两个公式一般不用，计算都是使用下面这个公式.
2.计算公式
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
标准差： D ( X ) \sqrt{D(X)} D(X)
3.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
3.常见分布的期望和方差
| 分布 | 期望 | 方差 |
|---|---|---|
| 0-1分布 | p p p | p ( 1 − p ) p(1-p) p(1−p) |
| 二项分布 | n p np np | n p ( 1 − p ) np(1-p) np(1−p) |
| 泊松分布 | λ \lambda λ | λ \lambda λ |
| 均匀分布 | a + b 2 \frac{a+b}2 2a+b | ( b − a ) 2 12 \frac{\left(b-a\right)^2}{12} 12(b−a)2 |
| 指数分布 | 1 λ \frac1\lambda λ1 | 1 λ 2 \frac1{\lambda^2} λ21 |
| 正态分布 | μ \mu μ | σ 2 \sigma^2 σ2 |
| 几何分布 | 1 p \frac1p p1 | 1 − p p 2 \frac{1-p}{p^2} p21−p |
| 瑞利分布 | π 2 σ \sqrt{\frac\pi2}\sigma 2πσ | σ 2 \sigma^2 σ2 |
4.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))]
1.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
2.协方差矩阵
二维随机变量 ( X 1 , X 2 ) (X_1,X_2) (X1,X2)有四个二阶中心矩（假设他们都存在），分别记为
C 11 = E { [ X 1 − E ( X 1 ) ] 2 } C 12 = E { [ X 1 − E ( X 1 ) ] [ X 2 − E ( X 2 ) ] } C 21 = E { [ X 2 − E ( X 2 ) ] [ X 1 − E ( X 1 ) ] } C 22 = E { [ X 2 − E ( X 2 ) ] 2 } C_{11}=E\left\{\left[X_1-E\left(X_1\right)\right]^2\right\}\\C_{12}=E\left\{\left[X_1-E\left(X_1\right)\right]\left[X_2-E\left(X_2\right)\right]\right\}\\C_{21}=E\left\{\left[X_2-E\left(X_2\right)\right]\left[X_1-E\left(X_1\right)\right]\right\}\\C_{22}=E\left\{\left[X2-E\left(X_2\right)\right]^2\right\} C11=E{[X1−E(X1)]2}C12=E{[X1−E(X1)][X2−E(X2)]}C21=E{[X2−E(X2)][X1−E(X1)]}C22=E{[X2−E(X2)]2}将他们排成矩阵的形式
( C 11 C 12 C 21 C 22 ) \begin{pmatrix}C_{11}&C_{12}\\C_{21}&C_{22}\end{pmatrix} (C11C21C12C22)这个矩阵称为随机变量 ( X 1 , X 2 ) (X_1,X_2) (X1,X2)的协方差矩阵.
设 n n n维随机变量 ( X 1 , X 2 , ⋯ , X n ) \left(X_1,X_2,\cdots,X_n\right) (X1,X2,⋯,Xn)的二阶混合中心距
C i j = C o v ( X i , X j ) = E { [ X i − E ( X i ) ] [ X j − E ( X j ) ] } , i , j = 1 , 2 , ⋯ , n C_{ij}=Cov\left(X_i,X_j\right)=E\left\{\left[X_i-E\left(X_i\right)\right]\left[X_j-E\left(X_j\right)\right]\right\},i,j=1,2,\cdots,n Cij=Cov(Xi,Xj)=E{[Xi−E(Xi)][Xj−E(Xj)]},i,j=1,2,⋯,n都存在，则称矩阵
C = ( C 11 C 12 ⋯ C 1 n C 21 C 22 ⋯ C 2 n ⋮ ⋮ ⋮ C n 1 C n 2 ⋯ C n n ) C=\begin{pmatrix}C_{11}&C_{12}&\cdots&C_{1n}\\C_{21}&C_{22}&\cdots&C_{2n}\\\vdots&\vdots&&\vdots\\C_{n1}&C_{n2}&\cdots&C_{nn}\end{pmatrix} C=⎝⎜⎜⎜⎛C11C21⋮Cn1C12C22⋮Cn2⋯⋯⋯C1nC2n⋮Cnn⎠⎟⎟⎟⎞为 n n n维随机变量 ( X 1 , X 2 , ⋯ , X n ) \left(X_1,X_2,\cdots,X_n\right) (X1,X2,⋯,Xn)的协方差矩阵.由于 C i j = C j i ( i ≠ j ; i , j = 1 , 2 , ⋯ , n ) C_{ij}=C_{ji}\left(i\neq j;i,j=1,2,\cdots,n\right) Cij=Cji(i=j;i,j=1,2,⋯,n)，因此上述矩阵是一个对称矩阵.
5.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。