9. Let A be a random variable with the following distribution:for k = 1,2,3Pr {A = k} =otherwiseDefine a random sequence {X;} as:• X1 = A,• for all i> 1:if X; = 1X;+1 =2if X; = 2X;+1 =3%3Dif X; = 3Xi+1 =1%3D(a) Is {X;}, Markov?(b) Define the random sequence {Y;}, according to:S1 if X; = 1Y; =2 otherwiseShow that {Y:} is not Markov.介介介

Answered: Pr{A = k} otherwise andom sequence…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Show that if X is not a deterministic random variable, then H(X) is strictly positive. What happens to the probabilities if a random variable is non-deterministic?
4. Suppose X0, X1, X2,... are iid Binomial (2, 5). If we view this sequence as a Markov chain with S = {0,1,2}, what is its PTM?
Consider a Markov chain in the set {1, 2, 3} with transition probabilities p12 = p23 = p31 = p, p13 = p32 = p21 = q = 1 − p, where 0 < p < 1. Determine whether the Markov chain is reversible.
A study of armed robbers yielded the approximate transition probability matrix shown below. The matrix gives the probability that a robber currents free, on probation, or in jail would, over a period of a year, make a transition to one of the states. То From Free Probation Jail Free 0.7 0.2 0.1 Probation 0.3 0.5 0.2 Jail 0.0 0.1 0.9 Assuming that transitions are recorded at the end of each one-year period: i) For a robber who is now free, what is the expected number of years before going to jail? ii) What proportion of time can a robber expect to spend in jail? [Note: You may consider maximum four transitions as equivalent to that of steady state if you like.]
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5. Explain what is meant by BLUE estimates and the Gauss-Markov theorem. Mathematics relevant proofs will be rewarded. [:
Which statements are true? Select one or more: a. Markov’s inequality is only useful if I am interested in that X is larger than its expectation. b. Chebyshev’s inequality gives better bounds than Markov’s inequality. c. Markov’s inequality is easier to use. d. One can prove Chebyshev’s inequality using Markov’s inequality with (X−E(X))2.
prove the property
Q3) The state transition matrix of a Markov random process is given by 1/3 1/3 1/6 1/6 5/9 0 0 4/9 2/5 1/5 1/5 1/5 0 0 3/20 17/20 [ ] Draw the state transition diagram and denote all the state transition probabilities on the same. Find P[X1 = 2] List the pairs of communicating states. Find P[X2 = 3| X1 = 2] Compute P[X2 = 2 | X0 = 1] Compute P[X3 = 3, X2 = 1, X1 = 2 | X0 = 3] (vii) Find P[X4 = 4, X3 = 3, X2 = 3, X1 = 1, X0 =2] where Xt denotes the state of the random process at time instant t. The initial probability distribution is given by X0 = [2/5 1/5 1/5 1/5].
1. Prove that Weiner Process is Markovian (Markov Process).
A path of length k in a Markov chain {X,,n= 0,1,...} is a sequence of states visited from step n to step n+k. Let pij be the transition probability from state i to state j. Show that starting from state i,, the probability that the chain follows a particular path i, - i„.1 - in2 - ... - i,t is given by %3D "x*\":= "X)d = i2.,Xpsk = inrk | X, = i,) n+1 n+l> n+2 n+k = P Pu.
4

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Pr{A = k} otherwise andom sequence {X;}, as: А, li>1: if X; = 1 Xi+1 if X; = 2 Xi+1 if X; = 3 Xi+1 }, Markov? e the random sequence {Y;} according to: 1 if X; = 1 2 otherwise Y; = that {Y;}, is not Markov.