A \"partially reflecting\" random walk on the states 1, 2, 3, 4 has the TM: P =

Question

A "partially reflecting" random walk on the states 1, 2, 3, 4 has the TM: P = 1 2 3 4[.2 .8 0 0 .2 0 .8 0 0 .2 0 .8 0 0 .2 .8]. (This is a regular transition matrix.) Suppose we want to find the mean number of transitions until the first visit to state 4, starting in state 1, 2, or 3. So we change state 4 to an absorbing state. Give the new transition matrix P for the resulting absorbing chain. (+4) Find the I - Q matrix for the absorbing chain of part (a). (+5) The inverse of the I - Q matrix in part (b) is: U = (I - Q)^-1 = [105/64 100/64 80/64 25/64 100/64 80/64 5/64 20/64 80/64]. Use this to find the mean number of transitions that the original random walk takes to first reach state 4, assuming that it starts in state 3. Give it as a simple fraction. (+4) Use the fundamental matrix given in part (c) to state the mean number of times that the original random walk visits state 2 before it first visits state 4, assuming that it starts in state 3. Give it as a simple fraction. Below are the sixth and seventh powers of P for the absorbing chain. Use them to find the probability that the original random walk will first reach state 4 in exactly 7 transitions, given that it starts in state 3. P^6 = 1 2 3 4[0.0269 0.0381 0.0829 0.8520 0.0095 0.0381 0.0174 0.9349 0.0052 0.0044 0.0174 0.9731 0.0000 0.0000 0.0000 1.0000]. P^7 = 1 2 3 4[0.0130 0.03810.0305 0.9183 0.0095 0.0111 0.0305 0.9488 0.0019 0.0076 0.0035 0.9870 0.0000 0.0000 0.0000 1.0000]. (+4)

Explanation / Answer

a) and b)

c) and d)

Mean number of transitions

e)

*The resolution reaches using Markov chains model, for each case.

0.2 0.16 0.64 0.04 0.32 0 0.04 0 0.16 50 -25 -200 -6.25 6.25 25 -12.5 6.25 56.25

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