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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 14EQ

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Let X be a random variable with sample space {1,2, 3} and probability distribu-
(G 1 ). Find a transition matrix P such that the Markov chain {X„}
tion T =
simulates X.

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12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction randomly choose which way to go. Figure 3.28 (a) Construct the transition matrix for the Markov chain that models this situation. (b) Suppose we start with 15 robots at each junction. Find the steady state distribution of robots. (Assume that it takes each robot the same amount of time to travel between two adjacent junctions.)
The transition matrix of a Markov chain is given by 0 0 P = (a) Find two distinct stationary distributions of this Markov chain. (b) Find the general form of the stationary distribution. (c) If 70) = (, 4, ¿, 1, ) is the initial probability vector at time 0, ,π(m) = (금' 등,을 ,). 4'4'6' 6' 6 then show that limn 12 HIN O -160 2/3 HIN O 230116 O 0 -16 0 116
How do you know that a stochastic process {Xn: n = 0, 1, 2, ...} with discrete states is a Markov chain? O The subscript n takes on only a finite number of values. O The number of states is finite. O None of the other choices O It is possible, though perhaps difficult, to construct a one-step transition matrix for it. « Previous Simpfun
Let X be a Poisson(A) random variable. By applying Markov's inequality to the random variable W = etx, t > 0, show that P(X ≥ m) ≤ e-tmex(et-1). Hence show that, for m > >, e-^ (ex) m P(X ≥ m) ≤ mm
Find the vector of stable probabilities for the Markov chain whose transition matrix is
Consider a binary choice model in random utility framework: U1 = Bo + B1x1 + · ·. + BrXk + u = x'B+u and Uo = 0, where Uj and Uo are utilities from y = 1 and y = 0. When u follows the standard logistic distribution [the CDF of u is A(u) 1+e« ], what is the partial effect from a1 on the choice probability of choosing 1? B1 A(x'B)B1 A(x'B)[1 – A(x'B)]B1
An individual can contract a particular disease with probability 0.17. A sick person will recover dur- ing any particular time period with probability 0.44 (in which case they will be considered healthy at the beginning of the next time period). Assume that people do not develop resistance, so that pre- vious sickness does not influence the chances of contracting the disease again. Model as a Markov chain, give transition matrix on your paper. Find the probability that a healthy individual will be sick after two time periods.
Consider a Markov chain {X, : n = 0, 1, - .-} on the state space S = {1,2,3,4} with the following transition matrix: 1/3 2/3 1/2 1/2 P = 1/4 3/4 1/4 1/4 1/2 Find Pr(X7 = 2|X1 = 3). %3D Determine the class(es) of the above Markov chain. Specify which state is recurrent and which state is transient. Justify your results.
Let a stochastic process (Xt) be given as follows: Xt = 3 + 0:5Xt-1 + t, where (t) is white noise with var ( t) = 1. a) What process is (Xt)? b) Explain briefly why this is a \conditional expectation" model, but not a \conditional variance" model. c) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for Xt+1. d) Suppose we observed Xt = 2:5, Xt-1 = 1:7. Compute a forecast for the variance of the process at time t + 1, that is, for the variance of Xt+1.
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"Random walk" is a term that refers to a commonly-used statistical model, of a journey consisting of N steps where all steps are the same length and the direction of each step is chosen randomly. Consider a 1-dimensional random walk, where each step (of length I) can go either forwards (+) or backwards (-). The statistical model is similar to a paramagnet, where each spin can be either up (+) or down (-). The equivalent of a “microstate" is defined by specifying whether each step in the walk is + or -. The equivalent of a "macrostate" is defined by specifying N. (how many of the N steps in the walk are +). The most probable macrostate is N. = N/2, meaning at the end of a long random walk you are most likely to end up back at your starting point. (a) The multiplicity function 2(N, N.) is very highly peaked around N. = N/2. Using the change-of-variable x = N. – N/2, find the multiplicity function 2(N, x) in the vicinity of the peak (x = 0). Your answer should have the form of a Gaussian.…
Let X be a Markov chain and let (nrr≥ 0} be an unbounded increasing sequence of positive integers. Show that Yr Xnr constitutes a (possibly inhomogeneous) Markov chain. Find the transition matrix of Y when nr = 2r and X is: (a) simple random walk, and (b) a branching process. =

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