Prof. Valerie R. Bencivenga Economics 329 November 6, 2013 MIDTERM EXAM #2 – ANSWERS 1. X1 = rate of return on bond fund 1 1 2 3 X2 = rate of return on bond fund 2 3 0.05 0.1 0.15 4 0.1 0.2 0.1 5 0.15 0.1 0.05 a. E[X1] = 1(0.3) + 2(0.4) + 3(0.3) = 2 b.       2 2 21Var(X ) (1 2) (0.3) (2 2) (0.4) (3 2) (0.3) 0.6 c.     31 2 i 1 i 2i1E[X |X 3] x P X x |X 3                    120.05 0.1 0.15E[X |X 3] 1 2 3 2.330.3 0.3 0.3         321 2 i 1 2 1 i 2i1Var[X |X 3] x E[X |X 3] P X x |X 3                       2 2 2120.05 0.1 0.15Var[X |X 3] (1 2.33) (2 2.33) (3 2.33) 0.5550.3 0.3 0.3 d. E[X2] = 3(.3) + 4(.4) + 5(.3) = 4 Cov(X1,X2) = (1 – 2)(3 – 4)(0.05) + (2 – 2)(3 – 4)(0.1) + (3 – 2)(3 – 4)(0.15) + (1 – 2)(4 – 4)(0.1) + (2 – 2)(4 – 4)(0.2) + (3 – 2)(4 – 4)(0.1) + (1 – 2)(5 – 4)(0.15) + (2 – 2)(5 – 4)(0.1) + (3 – 2)(5 – 4)(0.05) = 0.05 – 0.15 – 0.15 + 0.05 = – 0.2 e. X1 and X2 are dependent because the covariance is non-zero. (A non-zero covariance implies dependence, although a zero covariance does not imply independence.) Alternatively (for example), P(X1 = 1  X2 = 3) = 0.05  P(X1 = 1)P(X2 = 3) = (0.3)(0.3) = 0.09. f. We already have the means of both X1 and X2, the variance of X1, and the covariance between X1 and X2. The one remaining quantity we will need is the variance of X2.       2 2 22Var(X ) (3 4) (0.3) (4 4) (0.4) (5 4) (0.3) 0.6 Let G represent the gain in the value of the portfolio             1212XXG 1000 2000 10X 20X100 100            12221 2 1 2E[G] 10E[X ] 20E[X ] 10(2) 20(4) 100 (dollars)Var(G) 10 Var(X ) 20 Var(X ) 2(10)(20)Cov(X ,X )100(0.6) 400(0.6) 2(10)(20)( 0.2) 220 ($ squared)2. a. X = number of people with reservations who show up (assume 18 reservations per flight) X ~ b(n = 18, p = 0.8) At least one passenger will not have a seat if 16, 17, or 18 people with reservations show up       16 2 17 1 18 018 16 18 17 18 18P(X 16) C 0.8 0.2 C 0.8 0.2 C 0.8 0.20.172 0.081 0.018 0.271 b. There will be one or more empty seats if 14 or fewer people with reservations show up.            15 318 15P(X 14) 1 P(X 15) P(X 16)1 C 0.8 0.2 0.271 1 0.230 0.271 0.499 c.   E[X] np 18(0.8) 14.4        Var(X) np(1 p) 18(0.8)(0.2) 2.88 2.88 1.7 d. She will not get on the second flight if 15 or more people show up for the second flight, and similarly for the third flight. The probability that 15 or more people show up for any given flight is .501 (from part b). Since the flights are independent, the probability that 15 or more people show up for both the second and third flights is 20.501 0.251. e. Call profit Y   Y 1000 75X       E[Y] 1000 75E[X] 1000 75(14.4) 80 Yes, the airline is profitable 3. X = number of published editorial that are on education X ~ hypergeometric random variable with parameters N, S, and n N = 9 (number of editorials submitted) S = 4 (number of editorials in the population that deal with education) n = 3 (number of editorials published) --      S x N S n x 4 0 5 3N n 9 34! 5!5(4)C C C C4!0! 3!2!202P(X 0) 0.1199! 9(8)(7)C C 1683!6! 3(2) or 11.9%4. a. X = number of taxis demanded in a one-hour period X ~ Poisson (6)                           6i00 6 1 6 2 6 3 6 4 6 5 6 6 6P X 6 1 P X i6 e 6 e 6 e 6 e 6 e 6 e 6 e10! 1! 2! 3! 4! 5! 6!1 0.0025 0.0149 0.0446 0.0892 0.1339 0.1606 0.1606 0.3937 b. Now X = number of taxis demanded in a ½ -hour period X ~ Poisson (3)                    0 3 1 3 2 3 3 33i03 e 3 e 3 e 3 eP X 3 1 P X i 10! 1! 2! 3!1 0.0498 0.1494 0.2240 0.2240 0.3528 5. Xi = monthly income of migrant i  2iX ~ N 200, 50 a.        i180 200 X 220 200P 180 X 220 P50 50                          P 0.4 Z .4F 0.4 F 0.4 F 0.4 1 F 0.4 2F 0.4 1 2 0.655 10.311 b.  2iX , i 1,...,n i.i.d. N 200, 50 n = 25  250X ~ N 200,25         180 200 X 220 200P 180 X 220 P50 5025 n 25                          P 2 Z 2F 2 F 2 F 2 1 F 2 2F 2 1 2 0.9772 10.9544 c.            5 X 5P 5 X 5 P50 50n n n     P 1.645 Z 1.645 0.90        5501.645 5 1.645 n 16.45 n 27150nn6. a. X ~ N(10,42) Y ~ N(20, 122)                X 0 10P X 0 P P Z 2.5 1 F(2.5) 1 0.9938 0.00624                Y 0 20P Y 0 P P Z 1.67 1 F(1.67) 1 0.9525 0.047512 Not required – The second investment has a higher probability of a negative rate of return, even though its mean rate of return is higher, because of its …