大部分来自wiki
虽然想着英文好理解一些, 但是自己写还是会有好多用错词 的啊(
就当latex练习好了
a mathematical object defined as a family of random variables.
a stochastic process can also be interpreted as a random element in a function space.
If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.
(When interpreted as time, ) The index set has a finite or countable number of elements or not.
where each random variable takes values from.
( Ω , F , P ) (\Omega,F,P) (Ω,F,P)
where Ω \Omega Ω is a sample space, F F F is a σ \sigma σ-algebra, P P P is a probability measure.
( S , Σ ) (S,\Sigma) (S,Σ)
while S S S is the state space.
{ X ( t ) : t ∈ T } \{X(t):t\in{T}\} {X(t):t∈T}
while X ( t ) X(t) X(t) refer to the random variable with the index t t t
T T T is called the index set or parameter set.
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 } F_{t_1,t_2,\cdots,t_n}(x_1,x_2,\cdots,x_n) = P\{X(t_1)\leq x_1,X(t_2)\leq x_2,\cdots,X(t_n)\leq x_n\} Ft1,t2,⋯,tn(x1,x2,⋯,xn)=P{X(t1)≤x1,X(t2)≤x2,⋯,X(tn)≤xn}
If the distribution is independent,
P { X ( t 1 ) ≤ x 1 , X ( t 2 ) ≤ x 2 } = P { X ( t 1 ) ≤ x 1 } P { X ( t 2 ) ≤ x 2 } P\{X(t_1)\leq x_1,X(t_2)\leq x_2\}=P\{X(t_1)\leq x_1\}P\{X(t_2)\leq x_2\} P{X(t1)≤x1,X(t2)≤x2}=P{X(t1)≤x1}P{X(t2)≤x2}
m X ( t ) = E X ( t ) , t ∈ T m_X(t)=EX(t), t\in T mX(t)=EX(t),t∈T
B X ( s , t ) = E [ ( X ( s ) − m X ( s ) ) ( X ( t ) − m X ( t ) ) ] B_X(s,t) = E[(X(s)-m_X(s))(X(t)-m_X(t))] BX(s,t)=E[(X(s)−mX(s))(X(t)−mX(t))]
D X ( t ) = σ X 2 ( t ) = E [ ( X ( t ) − m X ( t ) ) 2 ] = E X 2 ( t ) − m X ( t ) 2 = E X 2 ( t ) − ( E X ( t ) ) 2 D_X(t)=\sigma^2_X(t) = E[(X(t)-m_X(t))^2] = EX^2(t)-m_X(t)^2 = EX^2(t)-(EX(t))^2 DX(t)=σX2(t)=E[(X(t)−mX(t))2]=EX2(t)−mX(t)2=EX2(t)−(EX(t))2
R X ( s , t ) = E [ X ( s ) X ( t ) ] R_X(s,t)=E[X(s)X(t)] RX(s,t)=E[X(s)X(t)]
while m X ( t ) m_X(t) mX(t) is the mean value of X ( t ) X(t) X(t), D X ( t ) D_X(t) DX(t) is the offset of X ( t ) X(t) X(t) to mean value at time t t t ,
B X ( s , t ) B_X(s,t ) BX(s,t)& R X ( s , t ) R_X(s,t) RX(s,t) represents the relevance of SP { X ( t ) , t ∈ T } \{X(t), t\in T\} {X(t),t∈T} from different time s , t s,t s,t .
D X = E X 2 − ( E X ) 2 DX=EX^2-(EX)^2 DX=EX2−(EX)2
if X ∼ U ( 0 , T ) X\sim U(0, T) X∼U(0,T):
E [ f ( X ) ] = 1 T ∫ 0 T f ( x ) d x E[f(X)]=\frac{1}{T}\int_{0}^{T}f(x)dx E[f(X)]=T1∫0Tf(x)dx
representing the mean value of every possible X in (0, T)
stochastic process { X ( t ) , t ∈ T } \{X(t), t\in T \} {X(t),t∈T} ,
if E X ( t ) = 0 EX(t) =0 EX(t)=0, and t 1 < t 2 ≤ t 3 < t 4 ∈ T : E [ ( X ( t 2 ) − X ( t 1 ) ) ( X ( t 4 ) − X ( t 3 ) ‾ ) ] = 0 t_1 \lt t_2 \leq t_3 \lt t_4 \in T : E[(X(t_2)-X(t_1))\overline{(X(t_4)-X(t_3)})]=0 t1<t2≤t3<t4∈T:E[(X(t2)−X(t1))(X(t4)−X(t3))]=0,
X ( t ) , t ∈ T {X(t), t\in T} X(t),t∈T is a process with orthogonal increments.
Specially, if T = [ a , ∞ ) T=[a, \infty) T=[a,∞) and X ( a ) = 0 X(a)=0 X(a)=0,
B X ( s , t ) = R X ( s , t ) = σ X 2 ( m i n ( s , t ) ) B_X(s,t)=R_X(s,t)=\sigma_X^2(min(s,t)) BX(s,t)=RX(s,t)=σX2(min(s,t))
N ( μ , σ 2 ) E X = μ , D X = σ 2 N(\mu,\sigma^2) \\EX=\mu, DX=\sigma^2 N(μ,σ2)EX=μ,DX=σ2
Specially, in standard Normal distribution,
μ = 0 , σ 2 = 1 \mu=0, \sigma^2=1 μ=0,σ2=1
It can be defined as a counting process, which represents the random number of events up to some time.
P { X ( t + s ) − X ( s ) = n } = e − λ t ( λ t ) n n ! P\{X(t+s)-X(s)=n\}=e^{-\lambda t} \frac{(\lambda t)^n}{n!} P{X(t+s)−X(s)=n}=e−λtn!(λt)n
let X ( t ) , t ≥ 0 {X(t), t \geq 0} X(t),t≥0 be a Poisson process, for $t,s \in [0,\infty) $ and s ≤ t s \le t s≤t ,
E [ X ( t ) − X ( s ) ] = D [ X ( t ) − X ( s ) ] = λ ( t − s ) E[X(t)-X(s)]=D[X(t)-X(s)]=\lambda(t-s) E[X(t)−X(s)]=D[X(t)−X(s)]=λ(t−s)
since X ( 0 ) = 0 X(0)=0 X(0)=0,
m X ( t ) = λ t σ x 2 ( t ) = λ t B X ( s , t ) = λ s m_X(t) = \lambda t \\ \sigma^2_x(t)=\lambda t \\ B_X(s,t)=\lambda s mX(t)=λtσx2(t)=λtBX(s,t)=λs
normally,
B X ( s , t ) = λ min ( s , t ) B_X(s,t)=\lambda\min(s,t) BX(s,t)=λmin(s,t)
P ( λ ) P ( X = k ) = λ k k ! e − λ E X = D X = λ P(\lambda) \\ P(X=k)=\frac{\lambda^k}{k!}e^{-\lambda} \\ EX = DX = \lambda P(λ)P(X=k)=k!λke−λEX=DX=λ
if { N ( t ) , t ≥ 0 } \{N(t), t \geq 0 \} {N(t),t≥0} is a Poisson process of λ \lambda λ,
{ Y k , k = 1 , 2 , ⋯   } \{Y_k, k=1,2,\cdots\} {Yk,k=1,2,⋯} is a set of independent and identically distributed random variables,
and is independent to { N ( t ) , t ≥ 0 } \{N(t), t \geq 0\} {N(t),t≥0},
X ( t ) = ∑ k = 1 N ( t ) Y k , t ≥ 0 , X(t)=\sum_{k=1}^{N(t)}Y_k,\ \ t\geq0, X(t)=k=1∑N(t)Yk, t≥0,
{ X ( t ) , t ≥ 0 } \{X(t),t\geq0\} {X(t),t≥0} is a Compound Poisson process.
E [ X ( t ) ] = λ t E ( Y 1 ) D [ X ( t ) ] = λ t E ( Y 1 ) 2 E[X(t)]=\lambda t E(Y_1) \\ D[X(t)]=\lambda t E(Y_1)^2 E[X(t)]=λtE(Y1)D[X(t)]=λtE(Y1)2
probability transition matrix as
P = [ p i j ] P = [p_{ij}] P=[pij]
Two-step transition probability matrix as
P ( 2 ) = P P P^{(2)}=PP P(2)=PP
assume state space I = { 1 , 2 , ⋯   , 9 } I=\{1,2,\cdots ,9\} I={1,2,⋯,9} ,
for state 1, the step T T T is the steps it takes to go back from state 1
for set { n : n ≥ 1 , p i i ( n ) > 0 } \{n:n\geq1,p_{ii}^{(n)}\gt0\} {n:n≥1,pii(n)>0},
d = d ( i ) = G . C . D { n : p i i ( n ) > 0 } d = d(i)=G.C.D\{n:p_{ii}^{(n)}>0\} d=d(i)=G.C.D{n:pii(n)>0}
d d d is the cycle of state i i i,
if d > 1 d\gt1 d>1, state i i i is periodic,
if d = 1 d=1 d=1, state i i i is aperiodic.
f i j = ∑ n = 1 ∞ f i j n f_{ij}=\sum_{n=1}^{\infty}f_{ij}^{n} fij=n=1∑∞fijn
f i j f_{ij} fij is the the probability that i can finally reach j,
when f i i = 1 f_{ii}=1 fii=1 , the state i i i is recurrent. The necessary and sufficient condition is
∑ n = 0 ∞ p i i ( n ) = ∞ \sum_{n=0}^{\infty}p_{ii}^{(n)}=\infty n=0∑∞pii(n)=∞
Specially, state i i i is ergodic state if it is aperiodic & recurrent.
{ π j = ∑ i ∈ I π i p i j , ∑ j ∈ I π j = 1 , π j ≥ 0 , \begin{cases} \pi_j=\sum_{i \in I}\pi_i p_{ij} ,\\\\ \sum_{j \in I}\pi_j =1, \pi_j \geq 0, \end{cases} ⎩⎪⎨⎪⎧πj=∑i∈Iπipij,∑j∈Iπj=1,πj≥0,
the expected time
μ i = 1 π i \mu_i=\frac{1}{\pi_i} μi=πi1
{ X ( t ) , t ≥ 0 } \{X(t),t\geq 0\} {X(t),t≥0} is a Birth Death process when
{ p i , i + 1 ( h ) = λ i h + o ( h ) , λ i > 0 , p i , i − 1 ( h ) = μ i h + o ( h ) , μ i > 0 , μ 0 = 0 , p i i ( h ) = 1 − λ i h − μ i h + o ( h ) , p i j ( h ) = o ( h ) , ∣ i − j ∣ ≥ 2 , \begin{cases} p_{i,i+1}(h)=\lambda_ih+o(h),&\lambda_i>0,\\ p_{i,i-1}(h)=\mu_ih+o(h),&\mu_i>0,\mu_0=0,\\ p_{ii}(h)=1-\lambda_ih-\mu_ih+o(h),\\ p_{ij}(h)=o(h), &|i-j|\geq2, \end{cases} ⎩⎪⎪⎪⎨⎪⎪⎪⎧pi,i+1(h)=λih+o(h),pi,i−1(h)=μih+o(h),pii(h)=1−λih−μih+o(h),pij(h)=o(h),λi>0,μi>0,μ0=0,∣i−j∣≥2,
p i j ′ ( t ) = λ j − 1 p i , j − 1 ( t ) − ( λ j + μ j ) p i j ( t ) + μ j + 1 p i , j + 1 ( t ) , i , j ∈ I p'_{ij}(t)=\lambda_{j-1}p_{i,j-1}(t)-(\lambda_j+\mu_j)p_{ij}(t)+\mu_{j+1}p_{i,j+1}(t),\ \ \ \ i,j \in I pij′(t)=λj−1pi,j−1(t)−(λj+μj)pij(t)+μj+1pi,j+1(t), i,j∈I
希望不会挂科吧。。