【機率論】3.2随機變量與多元随機變量及其機率分布

随機變量與多元随機變量及其機率分布

• 1.随機變量及其機率分布
• • 1.随機變量及機率分布
  • 2.分布函數的概念與性質
  • 3.離散型随機變量的機率分布
  • 4.連續型随機變量的機率密度
  • 5.常見分布
  • 6.随機變量函數的機率分布
  • 7.重要公式與結論
• 2.多元随機變量及其分布
• • 1.二維随機變量及其聯合分布
  • 2.二維離散型随機變量的分布
  • 3. 二維連續性随機變量的密度
  • 4.常見二維随機變量的聯合分布
  • 5.随機變量的獨立性和相關性
  • 6.兩個随機變量簡單函數的機率分布
  • 7.重要公式與結論

1.随機變量及其機率分布

1.随機變量及機率分布

  取值帶有随機性的變量，嚴格地說是定義在樣本空間上，取值于實數的函數稱為随機變量，機率分布通常指分布函數或分布律

2.分布函數的概念與性質

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

性質：(1) 0 ≤ F ( x ) ≤ 1 0 \leq F(x) \leq 1 0≤F(x)≤1

(2) F ( x ) F(x) F(x)單調不減

(3) 右連續 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.離散型随機變量的機率分布

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.連續型随機變量的機率密度

機率密度 f ( x ) f(x) f(x);非負可積，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為 f ( x ) f(x) f(x)的連續點，則: f ( x ) = F ′ ( x ) f(x) = F'(x) f(x)=F′(x)分布函數 F ( x ) = ∫ − ∞ x f ( t ) d t F(x) = \int_{- \infty}^{x}{f(t){dt}} F(x)=∫−∞x​f(t)dt

5.常見分布

(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) 二項分布: 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)=Cnk​pk(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!λk​e−λ,λ>0,k=0,1,2⋯

(4) 均勻分布 U ( a , b ) U(a,b) U(a,b)： f ( x ) = { 1 b − a , a < x < b 0 , f(x) = \{ \begin{matrix} & \frac{1}{b - a},a < x< b \\ & 0, \\ \end{matrix} f(x)={​b−a1​,a<x<b0,​

(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

​σ1​e−2σ2(x−μ)2​,σ>0,∞<x<+∞

(6)指數分布: E ( λ ) : f ( x ) = { λ e − λ x , x > 0 , λ > 0 0 , E(\lambda):f(x) =\{ \begin{matrix} & \lambda e^{-{λx}},x > 0,\lambda > 0 \\ & 0, \\ \end{matrix} E(λ):f(x)={​λe−λx,x>0,λ>00,​

(7)幾何分布: 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)超幾何分布: 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)=CNn​CMk​CN−Mn−k​​,k=0,1,⋯,min(n,M)

6.随機變量函數的機率分布

(1)離散型： 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)

則: 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​)=yi​​P(X=xi​)

(2)連續型： X   ~ f X ( x ) , Y = g ( x ) X\tilde{\ }f_{X}(x),Y = g(x) X ~fX​(x),Y=g(x)

則: 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)≤y​fx​(x)dx， f Y ( y ) = F Y ′ ( y ) f_{Y}(y) = F'_{Y}(y) fY​(y)=FY′​(y)

7.重要公式與結論

(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) 離散型随機變量的分布函數為階梯間斷函數；連續型随機變量的分布函數為連續函數，但不一定為處處可導函數。

(6) 存在既非離散也非連續型随機變量。

2.多元随機變量及其分布

1.二維随機變量及其聯合分布

  由兩個随機變量構成的随機向量 ( X , Y ) (X,Y) (X,Y)， 聯合分布為 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)

2.二維離散型随機變量的分布

(1) 聯合機率分布律 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) 邊緣分布律 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) 條件分布律 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⋅j​pij​​

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. 二維連續性随機變量的密度

(1) 聯合機率密度 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) 分布函數： 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) 邊緣機率密度： 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) 條件機率密度： 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.常見二維随機變量的聯合分布

(1) 二維均勻分布： ( 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) 二維正态分布： ( 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)=2πσ1​σ2​1−ρ2

​1​.exp{2(1−ρ2)−1​[σ12​(x−μ1​)2​−2ρσ1​σ2​(x−μ1​)(y−μ2​)​+σ22​(y−μ2​)2​]}

5.随機變量的獨立性和相關性

X X X和 Y Y Y的互相獨立: ⇔ 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​（離散型）

⇔ 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)（連續型）

X X X和 Y Y Y的相關性：

相關系數 ρ X Y = 0 \rho_{{XY}} = 0 ρXY​=0時，稱 X X X和 Y Y Y不相關，

否則稱 X X X和 Y Y Y相關

6.兩個随機變量簡單函數的機率分布

離散型： 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) 則：

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​)=zk​​P(X=xi​,Y=yj​)

連續型： ( 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)

則：

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)≤z​f(x,y)dxdy， f z ( z ) = F z ′ ( z ) f_{z}(z) = F'_{z}(z) fz​(z)=Fz′​(z)

7.重要公式與結論

(1) 邊緣密度公式： 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}=∬D​f(x,y)dxdy

(3) 若 ( X , Y ) (X,Y) (X,Y)服從二維正态分布 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)

則有：

1. 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與 Y Y Y互相獨立 ⇔ ρ = 0 \Leftrightarrow \rho = 0 ⇔ρ=0，即 X X X與 Y Y Y不相關。
3. 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) C1​X+C2​Y∼N(C1​μ1​+C2​μ2​,C12​σ12​+C22​σ22​+2C1​C2​σ1​σ2​ρ)
4.   X {\ X}  X關于 Y = y Y=y Y=y的條件分布為： 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​+ρσ2​σ1​​(y−μ2​),σ12​(1−ρ2))
5. Y Y Y關于 X = x X = x X=x的條件分布為： 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​+ρσ1​σ2​​(x−μ1​),σ22​(1−ρ2))

(4) 若 X X X與 Y Y Y獨立，且分别服從 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​),

則： ( 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}). C1​X+C2​Y ~N(C1​μ1​+C2​μ2​,C12​σ12​C22​σ22​).

(5) 若 X X X與 Y Y Y互相獨立， f ( x ) f\left( x \right) f(x)和 g ( x ) g\left( x \right) g(x)為連續函數， 則 f ( X ) f\left( X \right) f(X)和 g ( Y ) g(Y) g(Y)也互相獨立。