For each of the following definitions of the state X_n at time n (n=1,2,3...), determine whether the Markov property is satisfied.
1. X_n is a sequence of independent discrete random variables.
2. You have m distinct boxes, numbered 1 through m, each containing some tokens. On each token is written an integer from 1 to m. Each box contains at least one token, but different boxes may contain different numbers of tokens. A box may also contain multiple tokens with the same number. Assume that you know the distribution of tokens in each box.
At time 0, you pick one box at random, say box i. You pick one of the tokens in box i randomly (each token in the box is equally likely to be chosen), read the corresponding number (say j), and put the token back in box i. At the next time slot, you pick one of the tokens in box j randomly (each token in the box is equally likely to be chosen) and repeat this process forever. At time n, you will be choosing tokens from some box. Let X_n be the number of this box.
3. Alice and Bob take turns tossing a fair coin. Assume that tosses are independent. Whenever the result is Heads, Alice gives 1 dollar to Bob, and whenever it is Tails, Bob gives 1 dollar to Alice. Alice starts with A dollars and Bob starts with B dollars, for some positive integers A and B. They keep playing until one player goes broke. Let X_n be the amount of money that Alice has after the nth toss.
1. Yes
2. Yes
3. Yes
1. In this case, the Markov property is satisfied. Since the random variables are independent, the current state, X_n, contains all the information needed to determine the future states. The past states, X_1, X_2, ..., X_{n-1}, do not provide any additional information for predicting X_n.
2. In this case, the Markov property is satisfied. The current state, X_n, represents the box from which the next token will be chosen. Knowing the current state is sufficient to determine the future states. The past states, X_1, X_2, ..., X_{n-1}, are not needed to predict X_n.
3. In this case, the Markov property is not satisfied. The current state, X_n, represents the amount of money Alice has after the nth toss, but it does not provide enough information to predict the future states. The outcome of each coin toss is independent, but the amount of money each player has depends on the cumulative results of all the previous tosses. The past states, X_1, X_2, ..., X_{n-1}, are necessary to predict X_n.
1. For the sequence X_n to satisfy the Markov property, the probability distribution of X_n should depend only on the previous state X_{n-1} and not on any earlier states.
If X_n is a sequence of independent discrete random variables, then the Markov property is not satisfied. In this case, the probability distribution of X_n is completely independent of the previous state X_{n-1} and any earlier states. Each random variable X_n is independent of the previous states and follows its own probability distribution.
2. In the given scenario, the state X_n represents the number of the box from which a token is chosen at time n. If we know the distribution of tokens in each box, then X_n depends only on the previous state X_{n-1}. The Markov property is satisfied in this case.
At each time slot, the state X_n is determined solely by the previous state X_{n-1}. The state transition from X_{n-1} to X_n depends only on the number written on the token randomly chosen from the box X_{n-1}. The distribution of tokens within each box does not affect the transition, as the selection process is based on choosing a random token from the current box.
3. In this scenario, each player's amount of money after each toss depends on the previous state (i.e., the amount of money they had before the toss). Therefore, the Markov property is satisfied.
The state X_n represents the amount of money Alice has after the nth toss. Alice's money after each toss depends only on her previous amount of money and the outcome of the current toss, which is independent of earlier tosses. Each state transition from X_{n-1} to X_n is determined solely by the previous state X_{n-1} and the result of the nth coin toss. The Markov property holds in this case.
To determine whether the Markov property is satisfied for each of these definitions, we need to consider whether the future state, X_(n+1), depends only on the current state, X_n, and not on any previous states.
1. X_n is a sequence of independent discrete random variables.
In this case, each state is independent of the previous state, which means that the future state only depends on the current state. Therefore, the Markov property is satisfied.
2. You have m distinct boxes, numbered 1 through m. At each time slot, you randomly choose a token from a box, read the number on it, and put it back in the same box. Let X_n be the number of the box at time n.
In this scenario, the future state, X_(n+1), does not solely depend on the current state, X_n. It also depends on the result of the token selection process, which can be influenced by previous states. Therefore, the Markov property is not satisfied.
3. Alice and Bob take turns tossing a fair coin, and X_n represents the amount of money that Alice has after the nth toss.
In this case, the future state, X_(n+1), depends solely on the current state, X_n. The next state is determined only by the outcome of the current coin toss, independent of any previous states. Therefore, the Markov property is satisfied.
In summary:
1. Markov property is satisfied.
2. Markov property is not satisfied.
3. Markov property is satisfied.