Isye 2027

# Suppose y is not observed but that we wish to

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Unformatted text preview: ently rolled. Let Xk = 1 if one shows on the k th die 0 else Yk = 1 if two shows on the k th die 0 else. Let X = n=1 Xk , which is the number of one’s showing, and Y = n=1 Xk , which is the number k k of two’s showing. Note that if a histogram is made recording the number of occurrences of each of the six numbers, then X and Y are the heights of the ﬁrst two entries in the histogram. (a) Find E [X1 ] and Var(X1 ). (b) Find E [X ] and Var(X ). (c) Find Cov(Xi , Yj ) if 1 ≤ i, j ≤ n (Hint: Does it make a diﬀerence if i = j ?) (d) Find Cov(X, Y ). (e) Find the correlation coeﬃcient ρX,Y . Are X and Y positively correlated, uncorrelated, or negatively correlated? 4.8. MOMENTS OF JOINTLY DISTRIBUTED RANDOM VARIABLES 155 1 Solution (a) Each Xk is a Bernoulli random variable with parameter p = 6 , so E [Xk ] = 1 and 6 2 ] − E [X ]2 = p − p2 = p(1 − p) = 5 . Var(Xk ) = E [Xk k 36 (b) E [X ] = nE [X1 ] = n , and Var(X ) = nVar(X1 ) = 5n . 6 36 (c) If i = j then Cov(Xi , Xj ) = 0, because Xi and Yj are associated with diﬀerent, independent, dice rolls. If i = j the situation...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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