t3818m2ans - University of Colorado Department of Economics...

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- 1 - University of Colorado Department of Economics Economics 3818 Midterm Examination Prof. Jeffrey S. Zax 3 November 2005 Solutions (32) 1. X and Y have the following joint distribution: Y y 1 =2 y 2 =4 x 1 =2 .2 .1 Xx 2 =4 .1 .1 x 3 =6 .2 .3 (1) a. What is P(X=4, Y=4)? P(X=4, Y=4)=.1. (2) b. What is the marginal distribution of X? P(X=2)=.3, P(X=4)=.2, P(X=6)=.5. (3) c. What is P(Y=4|X=4)? () ( ) PY X X PX X == = =∩ = = = = 44 4 4 1 2 5 | , . . .. (4) d. Is P(Y=4)=P(Y=4|X=4)? Why or why not? What, if anything, does the relation- ship between these two probabilities tell you with regard to the question of whether X and Y are independent? .5=P(Y=4)=P(Y=4|X=4)=.5. This means that X and Y could be independent. But it does not prove that they are. A proof would require demonstrating that this relationship was true for all values of Y and X.
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- 2 - (2) e. What is the conditional distribution of X given that Y=2? P(Y=2) = P(Y=2,X=2)+P(Y=2,X=4)+P(Y=2,X=6) = .2 +.1 +.2 =.5. Therefore, () ( ) PX Y Y PY Y Y Y Y == = = = = = = 22 2 2 5 4 42 2 1 5 2 62 2 2 5 4 | , . . ., | , . . | , . . .. (3) f. What is E(X)? Using the answer to b., () () () EX xpx ii i = =++= = 1 3 23 42 65 44 ... . . (2) g. What is E(Y)? Using the answer to e., P(Y=4) = P(Y=4,X=2)+P(Y=4,X=4)+P(Y=4,X=6) = .1 +.1 +.3 =.5. Therefore, () () EY ypy i = =+= = 1 2 25 45 3 .
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t3818m2ans - University of Colorado Department of Economics...

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