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.
 Spring '08
 Zahrn
 The Land

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