100AHW3 - i,j(3 Please calculate the conditional...

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STAT 100A HWIII Due next Wed in class Problem 1: Suppose an urn has r red balls and b blue balls. We random pick a ball, and then we put three balls of the same color back to the urn. After that we randomly pick a ball again. (1) What is the probability that the second pick is red? (2) Optional: Suppose after the second pick, we again put three balls of the same color back to the urn, and then we randomly pick a ball again. What is the probability that the third pick is red? (The optional questions will be graded but will not be counted to the total score of your homework. They are just for fun.) Problem 2: Suppose a person does a random walk on three states 1, 2, and 3. At each step, he goes to one of the other two states with equal probabilities. Let X t be the state of this person at time t = 0 , 1 , ... , and X 0 = 1. (1) Please calculate the distribution of X t , t = 1 , 2 , 3 , 4 , 5. (2) Please calculate the conditional probability K ij = P ( X t +1 = j | X t = i ) for all pairs of (
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Unformatted text preview: i,j ). (3) Please calculate the conditional probability M ij = P ( X t +2 = j | X t = i ) for all pairs of ( i,j ). (4) Optional: Let K be the matrix whose ( i,j )-th element is K ij . Let M be the matrix whose ( i,j )-th element is M ij . Prove M = K 2 . (5) Please calculate P ( X 3 = i | X 4 = j ) for all pairs of ( i,j ). (6) Optional: Please calculate P ( X 3 = i | X 5 = j ) for all pairs of ( i,j ). (7) Please interpret the results in (1), (2), (3), (5) in terms of a population on the move. Problem 3: Suppose we flip a coin independently n times, and each time, the probability of a head is p . (1) Let X be the number of heads. Please explain that P ( X = k ) = ( n k ) p k (1-p ) n-k , where k = 0 , 1 ,...,n . (2) Let T be the number of flips until we get the first head. Please explain that P ( T = k ) = (1-p ) k-1 p , where k = 1 , 2 ,... 1...
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