Question

Roll a fair die repeatedly and let Y₁ ,Y₂, be the resulting numbers. Let Xn = | { Y₁,Y₂, . . . ,Yₙ }| be the number of values we have seen in the first n rolls for n > 1 and set Xo 0·Xₙ , is a Markov chain. (a) Find its transition probability. (b) Let T- minn : Xn = 6] be the number of trials we need to see all 6 numbers at least once. Find ET

Answer 1

The question pertains to determining the transition probabilities of a Markov chain related to dice rolls and calculating the expected number of trials to see all six faces of a die. Transition probabilities depend on how many unique outcomes have been observed, and ET can be found using the linearity of expectation for each step from observing i to i+1 unique outcomes.

Step-by-step explanation:

The student's question involves the concept of a Markov chain as applied to the process of rolling a fair die and determining transition probabilities as well as the expected number of trials needed to observe all six possible outcomes at least once.

(a) For a fair six-sided die, the transition probabilities of the Markov chain Xn depend on the current number of unique outcomes observed. If we have observed i unique numbers, the probability of observing a new unique number in the next roll is (6 - i)/6, while the probability of rolling a number already seen is i/6.

(b) To find ET, the expected number of trials to see all six numbers, we can use the idea of linearity of expectation and calculate the expected trials needed to go from i to i+1 unique numbers observed, summing these individual expectations for i from 0 to 5.