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 (1p ) nk , 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 ) = (1p ) k1 p , where k = 1 , 2 ,... 1...
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 Fall '09
 Wu
 Conditional Probability, Probability, Probability theory, conditional probability Kij, conditional probability Mij

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