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chapter2随机过程的基本概念

作者:互联网

这里写目录标题

分布函数的定义

对于给定的 [ X ( t 1 ) , X ( t 2 ) , . . . . . , X ( t n ) ] [X(t_1), X(t_2), .....,X(t_n)] [X(t1​),X(t2​),.....,X(tn​)]的联合分布函数:
F ( t 1 , t 2 , ⋯   , t n ; x 1 , x 2 , ⋯   , x n ) = P { X ( t 1 ) ≤ x 1 , X ( t 2 ) ≤ x 2 , ⋯   , X ( t n ) ≤ x n } \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:
F ≜ { F ( t 1 , t 2 , ⋯   , t n ; x 1 , x 2 , … , x n ) : t i ∈ T , x i ∈ R i , i = 1 , 2 , ⋯   , n , n > 0 } \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 X T X_T 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:
φ ( t 1 , t 2 , ⋯   , t n ; θ 1 , θ 2 , ⋯   , θ n ) = E { e i [ θ 1 X ( t 1 ) + ⋯ + θ n X ( t n ) ] } 称 为 : { φ ( t 1 , t 2 , ⋯   , t n ; θ 1 , θ 2 , ⋯   , θ n ) : t 1 , t 2 , ⋯   , t n ∈ T , n ≥ 1 } \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[θ1​X(t1​)+⋯+θn​X(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)=(1​t​)(YZ​)∼N(0,1+t2)(X(s),X(t))T=(11​st​)(YZ​)∼N(0,Σ), 其中 Σ=(1+s21+st​1+st1+t2​)

随机过程的数字特征

定义一

给定随机过程 X T = X ( t ) , t ∈ T X_T ={X(t), t \in T} XT​=X(t),t∈T,称
m ( t ) ≜ E [ X ( t ) ] = ∫ − ∞ + ∞ x d F ( t , x ) , t ∈ T 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 X T 的 均 值 函 数 X_T的均值函数 XT​的均值函数

定义二:

给定随机过程 X T = X ( t ) , t ∈ T X_T ={X(t), t \in T} XT​=X(t),t∈T,称
D ( t ) ≜ D [ X ( t ) ] = E [ X ( t ) − m ( t ) ] 2 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 X T X_T XT​的方差函数
 称  σ ( t ) = D ( t )  为过程  X T  的均方差函数.  \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 X T = X ( t ) , t ∈ T X_T ={X(t), t \in T} XT​=X(t),t∈T,称
C ( s , t ) ∧ Cov ⁡ ( X ( s ) , X ( t ) ) = E { [ X ( s ) − m ( s ) ] [ X ( t ) − m ( t ) ] } 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 X T X_T XT​的协方差函数
C ( s , t ) = E ( X ( t ) X ( s ) ) − m ( s ) m ( t ) D ( t ) = C ( t , t ) = E [ X ( t ) − m ( t ) ] 2 \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 X T = X ( t ) , t ∈ T , 称 R ( s , t ) ≜ E [ X ( s ) X ( t ) ] 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 X T X_T XT​的自相关函数

随机过程的分类

马尔可夫过程

标签:right,t2,chapter2,cdots,随机,array,t1,基本概念,left
来源: https://blog.csdn.net/qq_40552152/article/details/120988041