Markov chains. I am finding the continuous case a bit unintuitive. This problem concerns martingales and branching processes, but it is supposedly not hard. I just need a detailed solution to this problem, so that I can see what concepts are used to solve it.

A Markov chain in discrete time is irreducible, has state space I = { 0,1,2,?} and starts in 1. It is both a branching process and a martingale. Determine the probability of hitting 0.
Let {Xn , n = 0, 1, ..} with X0 =1 be a discrete time branching process. Here Xn denote the state of ...

Solution Summary

The solution shows that the probability of extinction in a Markov chain is certain if it is a branching process and martingale.

This problem involves the use of Matlab.
Step 1: Choose any MarkovChain with a 3x3 transition matrix that is irreducible and aperiodic. The reason it needs to be irreducible and aperiodic is because we are looking for a MarkovChain that converges. Calculate the stationary distribution of the MarkovChain by hand.
Ste

See attached file for better format.
A particle moves on the states 1, 2, 3, and 4 according to a time-homogeneous Markovchain {Xn ; n ≥ 0 }
with initial distribution p(0) = ( .2, .8, 0, 0 ) and transition matrix : p = [■(.6&.4&0&0@.6&0&.4&0@0&.6&0&.4@0&0&.6&.4)]
a) Draw the state diagram of the chain.

Please see attached file.
Consider the Markovchain with transition matrix (please see the attached file) where 0 <= p <= 1 and 0 <= q <= 1. Find an invariant distribution of this chain, For which values of p and q is the distribution unique? Given an initial distribution (please see the attached file), for which values of p

Category: Statistics > Simple Regression
Subject: birth-death process
Details: Let a Markovchain have transition probabilities Pi,i+1 = 1/(i+1) and Pi,0 = i/(i+1) for i=0,1,2, . . . .
i) Is the chain positive recurrent? Or transient or null recurrent?
ii) Does a stationary distribution exist? If it does, what is it?
S

Please see attached file.
Let a fair coin be tossed indefinitely. Denote respectively by H_n and T_n the total number of heads and tails after n tosses. Let X_n = H_n and Y_n = H_n - T_n. Are these Markovchains? If so, what is their transition matrix?

1. Suppose that shocks occur according to a Poisson process with rate A> 0. Also suppose that each shock independently causes the system to fail with probability 0 < p < 1. Let N denote the number of shocks that it takes for the system to fail and let T denote the time of the failure.
(a) FindP{T>tNrrn}.
(b) FindP{NrrnT=t}.
(

1. A production process contains a machine that deteriorates rapidly in both quantity and output under heavy usage, so it is inspected at the end of each day. Immediately after inspection the condition of the machine is noted and classified into one of four possible states:
State
0: good as new
1: Operable - minimum deteri

1. A bag contains 10 red, 12 green, and 8 yellow marbles. Assuming that all marbles are equally likely to be picked from the bag, what is the probability that the second marble is yellow, given that the first marble was yellow?
2. Given the following information, calculate the Predictive value positive and the Predic