Question

In a carnival game there is a pile of innumerous stuffed animals which come in six different shapes. When a game is played a stuffed animal is taken randomly from the pile and awarded. Let X0, X1, X2, X3, …. represent the total number of distinct animals that have been won up to and including games 0, 1, 2, 3, …. respectively.

Define the states in a MARKOV CHAIN to be {0, 1, 2, 3, 4, 5, 6}, representing the number of distinct animals won and let Pij be the (i, j)th element of the transition matrix P for the Markov Chain.

a) Determine the elements of the matrix (as fractions) use them to write out the matrix P. HINT: Should you want to label the rows/columns of your matrix start at zero and end at six!

b) Evaluate P22,3 by hand then determine the matrix P2 to confirm your result. THESE RESULTS SHOULD BE THE SAME.

c) Let E(Xn) represent the expected number of prizes after the nth game is played. Then En = (5/6)En-1 + 1

Verify this recursive equation by looking at the first row of the matrices P2, P3, P4 . Using the appropriate entries of those matrices determine the distributions of X2 , X3, X4, then calculate E(X2), E(X3), E(X4) using the above recursion equation

Answer 1

To determine the transition matrix P, we need to consider the probabilities of transitioning between each state. Each transition probability can be calculated based on the probability of winning a new distinct animal. The transition matrix P can then be constructed using these probabilities.

Step-by-step explanation:

To determine the elements of the transition matrix P, we need to consider the probabilities of transitioning between each state. In this case, there are 7 states representing the number of distinct animals won (0, 1, 2, 3, 4, 5, 6). The transition probabilities can be calculated based on the probability of winning a new distinct animal.

For example, let's consider the transition from state 0 to state 1. Since there are 6 different shapes of stuffed animals, the probability of winning a new distinct animal is 6/6 = 1. Therefore, P(0,1) = 1.

Similarly, the transition from state 1 to state 2 would have a probability of 5/6, since there are now 5 remaining distinct animals. Continuing this process, we can calculate all the transition probabilities and construct the transition matrix P.