这里写目录标题
分布函数的定义
对于给定的
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[X(t_1), X(t_2), .....,X(t_n)]
[X(t1),X(t2),.....,X(tn)]的联合分布函数:
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\begin{array}{l} F\left(t_{1}, t_{2}, \cdots, t_{n} ; x_{1}, x_{2}, \cdots, x_{n}\right)= \\ \quad P\left\{X\left(t_{1}\right) \leq x_{1}, X\left(t_{2}\right) \leq x_{2}, \cdots, X\left(t_{n}\right) \leq x_{n}\right\} \end{array}
F(t1,t2,⋯,tn;x1,x2,⋯,xn)=P{X(t1)≤x1,X(t2)≤x2,⋯,X(tn)≤xn}
称为过程的n维分布函数,记:
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\begin{aligned} F \triangleq &\left\{F\left(t_{1}, t_{2}, \cdots, t_{n} ; x_{1}, x_{2}, \ldots, x_{n}\right):\right.\\ &\left.t_{i} \in T, x_{i} \in R_{i}, i=1,2, \cdots, n, n>0\right\} \end{aligned}
F≜{F(t1,t2,⋯,tn;x1,x2,…,xn):ti∈T,xi∈Ri,i=1,2,⋯,n,n>0}
称F为
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XT的有限维分布函数族,在这个过程中,n维特征函数定义为:
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\begin{array}{l} \varphi\left(t_{1}, t_{2}, \cdots, t_{n} ; \theta_{1}, \theta_{2}, \cdots, \theta_{n}\right) \\ \quad=E\left\{e^{i\left[\theta_{1} X\left(t_{1}\right)+\cdots+\theta_{n} X\left(t_{n}\right)\right]}\right\} \end{array} \\ 称为: \begin{array}{r} \left\{\varphi\left(t_{1}, t_{2}, \cdots, t_{\mathrm{n}} ; \theta_{1}, \theta_{2}, \cdots, \theta_{n}\right):\right. \\ \left.t_{1}, t_{2}, \cdots, t_{n} \in T, n \geq 1\right\} \end{array}
φ(t1,t2,⋯,tn;θ1,θ2,⋯,θn)=E{ei[θ1X(t1)+⋯+θnX(tn)]}称为:{φ(t1,t2,⋯,tn;θ1,θ2,⋯,θn):t1,t2,⋯,tn∈T,n≥1}
随机过程的有限维分布函数满足一下的两个性质:
对称性: F ( t j 1 , ⋯ , t j n ; x j 1 , ⋯ , x j n ) = F ( t 1 , t 2 , ⋯ , t n ; x 1 , x 2 , . . , x n ) F\left(t_{j_{1}}, \cdots, t_{j_{n}} ; x_{j_{1}}, \cdots, x_{j_{n}}\right)=F\left(t_{1}, t_{2}, \cdots, t_{n} ; x_{1}, x_{2}, . ., x_{n}\right) F(tj1,⋯,tjn;xj1,⋯,xjn)=F(t1,t2,⋯,tn;x1,x2,..,xn)
相容性:对于任意固定的自然数m<n,均有:
F ( t 1 , t 2 , ⋯ , t m ; x 1 , x 2 , … , x m ) = F ( t 1 , t 2 , ⋯ , t m , ⋯ , t n ; x 1 , x 2 , … , x m , ∞ ⋯ ∞ ) = lim x m + 1 , , x n → ∞ F ( t 1 , t 2 , ⋯ , t n ; x 1 , … , x m , ⋯ x n ) \begin{array}{l} F\left(t_{1}, t_{2}, \cdots, t_{m} ; x_{1}, x_{2}, \ldots, x_{m}\right) \\ \quad=F\left(t_{1}, t_{2}, \cdots, t_{m}, \cdots, t_{n} ; x_{1}, x_{2}, \ldots, x_{m}, \infty \cdots \infty\right) \\ \quad=\quad \lim _{x_{m+1},, x_{n} \rightarrow \infty} F\left(t_{1}, t_{2}, \cdots, t_{n} ; x_{1}, \ldots, x_{m}, \cdots x_{n}\right) \end{array} F(t1,t2,⋯,tm;x1,x2,…,xm)=F(t1,t2,⋯,tm,⋯,tn;x1,x2,…,xm,∞⋯∞)=limxm+1,,xn→∞F(t1,t2,⋯,tn;x1,…,xm,⋯xn)注意:联合分布函数可以完全确定边缘分布函数!!!
题目:
设随机过程
X ( t ) = Y + Z t , t > 0 X(t)=Y+Z t, t>0 X(t)=Y+Zt,t>0
其中Y,Z相互独立,服从正态分布,求 X ( t ) X(t) X(t)的一,二维概率密度.
( Y z ) ∼ N ( 0 , I 2 ) X ( t ) = ( 1 t ) ( Y Z ) ∼ N ( 0 , 1 + t 2 ) ( X ( s ) , X ( t ) ) T = ( 1 s 1 t ) ( Y Z ) ∼ N ( 0 , Σ ) , 其中 Σ = ( 1 + s 2 1 + s t 1 + s t 1 + t 2 ) \left(\begin{array}{l} Y \\ z \end{array}\right) \sim N\left(0, I_{2}\right) \quad X(t)=\left(\begin{array}{ll} 1 & t \end{array}\right)\left(\begin{array}{l} Y \\ Z \end{array}\right) \sim N\left(0,1+t^{2}\right)\\ (X(s), X(t))^{T}=\left(\begin{array}{ll} 1 & s \\ 1 & t \end{array}\right)\left(\begin{array}{l} Y \\ Z \end{array}\right) \sim N(0, \Sigma) \text {, 其中 } \Sigma=\left(\begin{array}{cc} 1+s^{2} & 1+s t \\ 1+s t & 1+t^{2} \end{array}\right) (Yz)∼N(0,I2)X(t)=(1t)(YZ)∼N(0,1+t2)(X(s),X(t))T=(11st)(YZ)∼N(0,Σ), 其中 Σ=(1+s21+st1+st1+t2)
随机过程的数字特征
定义一
给定随机过程
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X_T ={X(t), t \in T}
XT=X(t),t∈T,称
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m(t) \triangleq E[X(t)]=\int_{-\infty}^{+\infty} x d F(t, x), \quad t \in T
m(t)≜E[X(t)]=∫−∞+∞xdF(t,x),t∈T
为过程
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X_T的均值函数
XT的均值函数
定义二:
给定随机过程
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X_T ={X(t), t \in T}
XT=X(t),t∈T,称
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D(t) \triangleq D[X(t)]=E[X(t)-m(t)]^{2}
D(t)≜D[X(t)]=E[X(t)−m(t)]2
为过程
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\text { 称 } \sigma(t)=\sqrt{D(t)} \text { 为过程 } X_{T} \text { 的均方差函数. }
称 σ(t)=D(t)
为过程 XT 的均方差函数.
定义三:
给定随机过程
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X_T ={X(t), t \in T}
XT=X(t),t∈T,称
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C(s, t)^{\wedge} \operatorname{Cov}(X(s), X(t))=E\{[X(s)-m(s)][X(t)-m(t)]\}
C(s,t)∧Cov(X(s),X(t))=E{[X(s)−m(s)][X(t)−m(t)]}
为过程
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XT的协方差函数
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\begin{array}{c} C(s, t)=E(X(t) X(s))-m(s) m(t) \\ D(t)=C(t, t)=E[X(t)-m(t)]^{2} \end{array}
C(s,t)=E(X(t)X(s))−m(s)m(t)D(t)=C(t,t)=E[X(t)−m(t)]2
定义四
给定随机过程
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X_T= {X(t), t \in T},称 R(s, t) \triangleq E[X(s) X(t)]
XT=X(t),t∈T,称R(s,t)≜E[X(s)X(t)]
为过程
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XT的自相关函数