Let N≥2 be an integer. We can consider {0,1,⋯,N−1} to be a "circle" by assuming that N−1 is adjacent to 0 as well N−2. Let Xn be simple random walk on the circle. The transition probabilities are pk,k−1=pk−1,k=0.5,k=1,⋯,N−1,p0,N−1=pN−1,0=0.5. Let N=6, (a) (10 points) What is the transition matrix P ? (b) (10 points) Is there a limiting probability vector? If yes, what is it? If no, what is the period? (c) (10 points) Is there any invariant probability distribution? If yes, what is it? Is it unique? If no, why?