Suppose that X k is a time-homogenous Markov chain. Show thatP{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
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Suppose that X k is a time-homogenous Markov chain. Show that
P{X3= j3, X2= j2|X0= j0,X1 = j1}= P{X3 = j3 | X2 = j2} P{X2 = j2|X1 = j1}.
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- Consider the following model to grow simple networks. At time t = 1 we start with a complete network with no = 6 nodes. At each time step t> 1 a new node is added to the network. The node arrives together with m = 2 new links, which are connected to m = 2 different nodes already present in the network. The probability II, that a new link is connected to node i is: ki II¿ = Z - 1 N(t-1) with Z(k-1) - j=1 where k, is the degree of node i, and N(t 1) is the number of nodes in the network at time t-1. (b) What is the average node degree (k) at time t? What is the average node degree in the limit t→ ∞o?Find the vector of stable probabilities for the Markov chain whose n matrix is 0.2 0.4 0.4 1 1 [ WHelp me fast so that I will give Upvote.
- why is the covariance of a deterministic and a stochastic process 0? This relats to Arithmetic Bronian MotionCan someone please help me with this question. I am having so much trouble.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. Let X be a Markov chain with state space S = {1, 2, 3) and transition matrix (² where 0 < p < 1. Prove that P = 0 Ph = P P P 1-P 0 0 P 1-p ain a2n a3n aln a2n a3n where ain + wa2n + w²a3n = (1 − p + pw)", w being a complex cube root of 1. a3n a2n ain3. suppose that the manufacturer keeps a spare machine that only is used when the primary machine is being repaired. During a repair day, the spare machine has a probability of 0.1 of breaking down, in which case it is repaired the next day. Denote the state of the system by (x, y), where x and y, respectively, take on the values 1 or 0 depending upon whether the primary machine (x) and the spare machine (y) are operational (value of 1) or not operational (value of 0) at the end of the day. [Hint: Note that (0, 0) is not a possible state.] (a) Construct the (one-step) transition matrix for this Markov chain. (b) Find the expected recurrence time for the state (1, 0).Suppose qo = [1/4, 3/4] is the initial state distribution for a Markov process with the following transition matrix: %3D [1/2 1/2] M = [3/4 1/4] (a) Find q1, 92, and q3. (b) Find the vector v that qo M" approaches. (c) Find the matrix that M" approaches.
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