A standard 6-sided die is rolled repeatedly. Let

Xi be the number of 6s seen in the first i rolls. Let Yi be the largest number seen in the first i rolls.

a) Explain briefly why X1,X2,X3, . . . is a Markov chain and give the transition probabilities and the transition graph.

b) Explain briefly why Y1,Y2,Y3, . . . is a Markov chain and give the transition matrix.

The die is modified so that it behaves normally on the first roll but threafter it can never produce the same number twice in succession (with the other 5 possibilities being equally likely). The random variables Xi and Yi are defined as before.

c) Is X1,X2,X3, . . . a Markov chain?

d) Is Y1,Y2,Y3 . . . a Markov chain?

Can you help me with part c and d please?