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103 transition-matrix questions.

I just read on wikipedia that a way to check whether a Markov chain is ergodic is to compute the eigenvalues of the transition matrix, and if those are all (except for 1) less than 1, then the chain ...

For a four state $(s_1, s_2, s_3, s_4)$ Markov chain, the transition probability matrix is given by: $$P = \begin{bmatrix} 1-a & a & 0 & 0\\ 1-b & 0 & b & 0\\ 1-c & 0 &...

I am currently trying to understand a proof for the above, which states that, in other words, there exists a unique $\overrightarrow{v}$, such that $\overrightarrow{v}P = \overrightarrow{v}$ for the ...

I've been given the following definition: For a THMC with one step transition matrix $\mathbf{P}$, the row vector $\mathbf{\pi}$ with elements $(\pi_{i})_{i \in S}$ (where $S$ is the state space) ...

Is there any shortcut to find $P^{n}=\begin{pmatrix} 1-p & p \\ q & 1-q \end{pmatrix}^n$ quickly and elegantly? This type of matrix often comes up while dealing with Markov chains. ...

I need your help please, we have the system X(t+1)= \begin{bmatrix} -1 & \frac{2+(-1)^{t}}{2} \\ \frac{2+(-1)^{t}}{2} & -1 \end{bmatrix} X(t) By using this formula $\phi(t)=A(t-1)...A(1)A(0)$...

I have the following exercise and I have some doubts about it's solution. a) Find P{$X_4$=1} b) Calculate the limiting distribution. c) What is the long run rate of repairs per unit time? My ...

Let P be a transition matrix of a Markov Chain. Let $p_{ij}^{n}=P\left [ X_{n}=j | x_{0}=i\right ]$ be the transition probability of a Markov Chain from an initial state i to a final state j in n-...

Let $\xi_n$, $n \in \mathbb{Z}_+$ be a sequence of i.i.d random variables over $\mathbb{R}$ with the density $p(x)$. Consider the sequence $$ \eta_n := \sum\limits_{k=1}^n\left(a\xi_k + b\xi_{k + 2}\...

Is there any direct proof to show that if eigenvalues of an LTI system are negative then the transition matrix ( or matrix exponential with respect to time) e^{At} decays to zero when t goes to ...

I am curious if an autoregressive process of order $k:$ $X_{t}= c+ \sum_{i=1}^{k}\phi_i X_{t-i} + \epsilon_i$ can be expressed as a $k$-step Markov chain with transition probability $$ P_{ij}^{k} = ...

$\mathbf{Theorem}$: Suppose $X$ is an irreducible Markov chain with transition matrix $P$. Let $\lambda$ be an invariant measure for $P$, i.e. $\lambda P = \lambda$. Suppose that some $\lambda_k > ...

Suppose that $P$ is the transition matrix of an irreducible, recurrent markov chain with countable state space $I$. Let $\pi$ be an invariant measure on the chain, i.e. $\pi = \pi P$. Given that $\...

I got an exercise in matlab were i need to solve two things. I think i have solved the first part but struggle to solve part 2. Info: A small version of the game Snakes and Ladders is shown in the ...

For some reason I have been struggling with this problem for the past couple hours. I believe I have solved part a. Since there are 6 states (assuming a standard die and the die is fair), then there ...

I am struggling with part B of this problem. I understand Markov chains and transition matrices but I'm stuck on where to start. Maybe it is just the wording of the problem. Can anybody point me in ...

Context of Question: "Algorithms for Reinforcement Learning", Csaba Szepesvari pg 11 Excerpt: In the case of the inventory control problem, the MDP was conveniently speci ed by a transition ...

Given v$_1=\left( \begin{array}{ccc} 3\\ -4 \end{array} \right)$, v$_2=\left( \begin{array}{ccc} 2\\ 5 \end{array} \right)$, $S=\left( \begin{array}{ccc} -1 & 7\\ 2 &-5 \end{array} \right)$ ...

As the title suggests, I am trying to understand how to compute a $n$-step transition probability given a transition probability matrix. Please understand that this is purely for me to prepare for ...

I am looking at Markov switching GARCH process. Intrinsically at each time instant I am choosing the conditional volatility depending on the state $\Delta_n$ of an underlying regime which follows a ...

Let $S=[1 \;1\;1],[1 \;2\;3],[1 \;0\;1]$ and $T=[0 \;1\;1],[1 \;0\;0],[1 \;0\;1]$. Find the transition matrix $P_{S\leftarrow T}$ from the set of ordered basis T to the set of ordered basis S. All ...

Let $(X_t)_{t\in \mathbb Z}$ be a homogeneous Markov chain on some state space $S$ with transition prbabilities $P=[P_{ij}]_{i,j\in S}$. The law $p_i(t):=\mathbb P(X_{t}=i)$ of $X_t$ satisfies: \begin{...

I'm stuck on this question because $PJ<---B$ isn't given. If it helps the previous question asked to find $x_G$ which I found to be $<-2,-11,-17>$. I would really appreciate your help. ...

(i'd really appreciate if someone fixed the notation, i have no clue how to do it. thanks) You are given this information at the start of the question: Really stuck on this, i have no clue what to ...

I am looking for the transition probabilities of a two-state ($0,1$) Markov chain $N(t)$: $P_{00}(t):=$Prob$(N(t)=0|N(0)=0),$ $P_{10}(t):=$Prob$(N(t)=0|N(0)=1).$ In more details, my setup has a ...

Hi I have some problem understanding the calculation of the following transition values for my transition matrix. The task is the following: A system consists of two elements that both work ...

I have a Markov chain, whose transition intensity matrix (or 'generator matrix') is: I know that, to get from this to the transition probability matrix $\textbf{P(t)}$ (where $t$ is time), I need to ...

If given a transition matrix, is there an easy and efficient method to determine if this matrix is absorbing, ergodic, and/or regular? I understand the definitions of all three terms, but if given a $...

Let $(Y_i)_{i\geq0}$ be i.i.d. random variables, following the uniform distribution in state space S = $\{1,...,6\}$. The goal is to show that the following sequence of random variables is a Markov ...

Let be the following framework : transition model : There is an initial state noted start and two final states in (4.3) and (4.2). The desired action is performed with a probability of 0.8, but with ...

Consider a population of individuals subject to a birth and death process. Assume that each organism dies in $(t,t+h]$ with probability $\mu(h)+o(h)$, it generates a descendant in $(t,t+h]$ with ...

The problem goes as follows: $$ P=\left( \begin{matrix} a & 0.6\\ 1-a & 0.4\\ \end{matrix} \right) $$ Determine the value of the parameter $a \in [0,1]$ for which $P$ does ...

12/15/2017 - 12:53 PM US EDST Please note: as a result of the decision by mercio, Math_QED, user8795, HK Lee, kjetil, and b halvorsen to put the question on hold because it is unclear to them, I have ...

The states of a system are: (1) Initialisation (2) Main Menu (3) Termination (4) Failure Probability p1 of failure and q1 of moving to the main menu from initialisation where p1 + q1 = 1 Probability ...

Let $$\pmatrix{\dot x_1\\ \dot x_2\\ \dot x_3} = \underbrace{\pmatrix{0 & 3 & 2\cos(7t)\\-3 & 0 & -2\sin(7t)\\-2\cos(7t) & 2\sin(7t) & 0}}_{A(t)}\underbrace{\pmatrix{x_1(t)\\...

Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn contains i white ...

Let's say I have a discrete time, time-homogeneous Markov chain $X = \{X_{1}, \dots , X_{n}\}$ with state space $S= \{1,2,3\}$ and a transition matrix: \begin{bmatrix} .4 & .3 & .3\\ .3 &...

As the title says I have to answer the following: Identify integers $1 \leq n /leq 20$ where $p_{0,0}^{(n)} > 0$ by using a transition matrix. I know this is with the states {0,1,2,3,4,5,6,7} and ...

I m dealing with the Kalman filter and the Signal Process is given by X(n+1) = B*X(n) + C*W(n+1)with W(n) ∈ R^n is i.i.d standard normal distributed, B ...

I am confusing myself with a problem which I give below. Can someone please comment and point out my mistakes. Say I have a two state system $S_1$ and $S_2$ the probability to transition to the two ...

Suppose we have a linear map $T$ whose matrix is $A$ such that $$A: V \rightarrow V$$ where $A$ is the transition matrix from the basis $v_1,v_2,...,v_n$ to the basis $w_1,w_2,...,w_n$. If the ...

I am working through Durrett's book Essentials of Stochastic Processes, which is quite good. He had a problem in the first chapter on Markov processes, and I had trouble understanding how he obtained ...

I have a transition matrix $$\mathbf P = \begin{bmatrix}0.9 & 0.06 & 0.04 \\ 0.6 & 0.3 & 0.1\\0.7 & 0.25 & 0.05\end{bmatrix}$$ The different states are labeled A, I, O in ...

Given the Markov transition matrix $$P=\left( \begin{array}{ccccccc} 0 & 0 & 1 & 0 & 0\\ 0 & 0.3 & 0 & 0 & 0.7\\ 1 & 0 & 0 & 0 & 0\\ 0.4 & 0 & ...

Suppose there exists an ergodic Markov chain with symmetric transition probabilities. For this Markov chain, why is the stationary distribution uniform?

This is a lemma for the Theorem of the Inverse of a Transition Matrix on p.211 of Larson's Elementary Linear Algebra 8e. Let $B=(v_1,...,v_n), B' =(u_1,...,u_n)$ be bases of a vector space $V$. If $...

Suppose that: Coin $1$ has probability of $0.7$ of coming up heads Coin $2$ has probability of $0.6$ of coming up heads If the coin flipped today comes up: heads: then we select coin $1$ to flip ...

I have a problem that need to calculate the average. 10 cities are arranged in a line. 2 cities are connected to two bridges, one is a good bridge, one a bad bridge. Passing the bad bridge will ...

A is a Regular Transition Matrix $\Rightarrow$ $\lim\limits_{m \to \infty} A^m$ exists and rank 1 At the above proposition, what does "regular" mean?

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