# MT 2 Spring 2016 answers - Dr Valerie R Bencivenga...

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Dr. Valerie R. Bencivenga April 5, 2016 Economics 329 Spring 2016 MIDTERM EXAM #2: RIGHT ANSWERS 1. a.   E X 0 .1 1 .2 2 .4 3 .2 4 .1 2 (5 points) b.             2 2 2 2 2 Var X 0 2 .1 1 2 .2 2 2 .4 3 2 .2 4 2 .1 1.2 (5 points) c. Y 100X 10(4 X) Y 110X 40 (5 points)       E Y 110E X 40 110 2 40 180 (5 points)     2 2 Var Y 110 Var X 110 1.2 14,520 (5 points) 2. W 40X Y (5 points)     E W 40E X E Y 40 100 1000 3000 (5 points)    2 2 Var W 40 Var X 1 Var Y 2 40 1 Cov X,Y 1600 81 1600 80 252 111,040   (5 points) 3. NUMBER OF BEDROOMS (X) 3 4 5 NUMBER OF LIVING AREAS (Y) 1 0.2 0.05 0.05 0.3 2 0.05 0.3 0.05 0.4 3 0.05 0.05 0.2 0.3 0.3 0.4 0.3 Students do not need to include the marginal probability distributions of X and Y for a full- credit answer. They’ll use them in the answers to parts a and c, though. Material in blue boxes is no required for a correct answer.

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a.   E X 3 0.3 4 0.4 5 0.3 4 (5 points) b. 0.05 0.05 0.2 0.15 0.2 1 1.35 E X|Y 3 3 4 5 4.5 0.3 0.3 0.3 0.3 0.3 (5 points) c. We need E[Y] in order to compute the covariance:   E X 1 0.3 2 0.4 3 0.3 2 Here is the covariance, with all of terms included. This is a correct answer but it is not required:                   Cov X,Y 3 4 1 2 0.2 4 4 1 2 0.05 5 4 1 2 0.05 3 4 2 2 0.05 4 4 2 2 0.3 5 4 2 2 0.05 3 4 3 2 0.05 4 4 3 2 0.05 5 4 3 2 0.2 Here is the covariance, calculated from the non-zero terms only:         Cov X,Y 3 4 1 2 0.2 5 4 1 2 0.05 3 4 3 2 0.05 5 4 3 2 0.2 0.2 0.05 0.05 0.2 0.3 (5 points) d. X and Y are not independent. Cov X,Y 0 implies X and Y are dependent. (5 points) Alternatively,   E X|Y 3 4.5 E X 4 implies dependence. (It is sufficient to point out that any of the conditional probabilities does not equal the marginal probability.) Alternatively,    P X 3 Y 1 0.2 P X 3 P Y 1 0.3 0.3 0.09 , and therefore X and Y are dependent. (It is sufficient to point out that any of the joint probabilities does not equal the product of the marginal probabilities.)
4.a. N = 30 (size of the population), S = 5 (number of successes in the population), n = 5 (number of trials) (3 points)

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