ECON 705 Midterm: 2012-705-Fall Econometrics Prelim Exam 2
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Friday august 17, 2012, time limit: 150 minutes. Instructions: (i) the exam consists of two parts. Let y = exp(x). (i) (4 points) suppose that x (cid:24) n ((cid:22); (cid:27)2). Compute the expected value e[y ]. (ii) (3 points) suppose that x (cid:24) n ((cid:22); (cid:27)2). Derive the proba- bility density function of y . (iii) (4 points) suppose that x1 and x2 are independent and have n (0; 1) distri- butions. De(cid:133)ne y1 = x1 + x2 and y2 = x1 (cid:0) x2. Use a change-of-variables argument to obtain the joint probability density for (y1; y2). Then compute the pdf for the marginal distribution of y1. Consider the linear regression model yi = xi(cid:18) + ui; uijxi (cid:24) iid(0; 1); xi (cid:21) (cid:15) > 0; e[x2 i ] = q; i = 1; : : : ; n. Moreover, the xi(cid:146)s are also independent across i.
