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fexamS-f08 - Economics 506 FINAL EXAM SOLUTION Name_Ali...

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Economics 506 FALL 2008 FINAL EXAM SOLUTION Name: __Ali Toossi_________________________ e- mail:_______________________ 1) [14]. 40% of all shoppers at Schnucks are men. 70% of men shoppers buy Pepsi. 20% of women shoppers buy Pepsi. What is the probability that a shopper is a man given they buy Pepsi? P (Pep |M) = .7 P (M)= .4 therefore P (Pep and M) = .28 P (Pep |W) = .20 P (W) = .6 therefore P (Pep and W) = .12 P (Pep and M) + P (Pep and W) = P (Pep) = .4 P (M |Pep) = P (Pep and M) / P (Pep) = .28/.4 = .7 2)[20] A discrete bivariate distribution is the following: Y -1 +1 0 | .3 .1 X 1 | .2 0 2 | 0 .4 Is this an appropriate probability distribution? Yes. Sum of all probabilities is one. Find marginal distribution of X. X: 0, 1, 2 P (x): 0.4, 0.2, 0.4 Find the mean, variance, and standard deviation of X. E (X) = 1, E (X 2 ) = 1.8, V (X) = 0.8, σ x = 0.894427 Find the conditional distribution of X given Y = -1. X: 0, 1, 2 P (X|Y=-1): 3/5, 2/5, 0 (Use the back side if you need to) 1
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Economics 506 FALL 2008 FINAL EXAM Find the mean and variance of X given Y = -1 E (X|Y=-1) = 2/5 E (X 2 |Y=-1) = 2/5 V (X|Y=-1) = 6/25 Find the covariance of X and Y. XY: 0, -1, -2, 0, 1, 2 P (X, Y): 0.3, 0.2, 0, 0.1, 0, 0.4 E (XY) = 0.6 E (X) =1, E (Y) = 0 COV (X, Y) = E (XY)-E (X) E (Y) = 0.6 Find the correlation coefficient of X and Y. V (Y) = 1 ρ =COV (X, Y)/ σ x σ y = 0.6/(0.894427*1) = 0.67 3)[12] X and Y have a joint bivariate normal distribution with the following parameters: μ x = -2, μ y = + 12, σ x = 2, σ y = 5, ρ = -0.6 Specify the form and the parameters of the marginal distribution of Y. Y ~ N (12, 25) Specify the form and the parameters of the conditional distribution of (Y | X = 4). Y|X=4 ~ N {12-0.6(5/2)(4+2)=3, 25(1-0.36) =16} (Use the back side if you need to) 2
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Economics 506 FALL 2008 FINAL EXAM 4)[20] X and Y have a joint bivariate distribution.
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