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Homework1Key - Economics 4261 Introduction to Econometrics...

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Economics 4261 Introduction to Econometrics Fall 2010 HOMEWORK 1 : MATH, STATS, AND PROBABILITY ANSWER KEY Question 1 Let X be a random variable with pdf f defined by f p x q : C p 5 ¡ x 2 q for ¡ 1 ¤ x ¤ 2 and f p x q : 0 otherwise. (a) What is the value of constant C ? (b) What is the cdf of X ? (c) Calculate E r X s , var r X s , and the standard deviation (SD) of X . .................................................................................................................. (a) The cdf must integrate to unity, where the cdf is given by F p x q : » C p 5 ¡ x 2 q dx. Hence, we must calculate C such that 1 » 2 ¡ 1 C p 5 ¡ x 2 q dx. This is the same as solving 1 C ¤ ¢ 5 x ¡ x 3 3 x 2 ¡ C ¤ ¢ 5 x ¡ x 3 3 x ¡ 1 C ¤ 12 . Hence, C 1 { 12 is the value we were looking for. (b) The cdf is given by: F p x q 5 12 x ¡ x 3 36   14 36 , for ¡ 1 ¤ x ¤ 2 . 1 (c) (i) First, E r X s ³ X xf p x q dx . For this problem, E r X s » 2 ¡ 1 1 12 ¤ x ¤ p 5 ¡ x 2 q After integrating, the answer is obtained from calculating ¢ ¡ 1 48 x 2 p x 2 ¡ 10 q x 2 ¡ ¢ ¡ 1 48 x 2 p x 2 ¡ 10 q x ¡ 1 The answer is E r X s 5 { 16. (ii) Next we calculate var p X q , where we use the important identity var p X q E r X 2 E r X s 2 . We already know the second term since E r X s 3 { 16. The first term is given by E r X 2 s ³ 2 ¡ 1 x 2 f p x q dx . Our calculation is: var p X q ¢ 5 p 2 q 3 36 ¡ p 2 q 5 60 ¡ ¢ 5 1 q 3 36 ¡ 1 q 5 60 ¡ ¢ 5 16 2 Or rather, var p X q 771 { 1280. (iii) Finally, SD p X q is simply the square root of the variance of X . The answer is b 771 1280 . 1 Remember to subtract F 1 q ! kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 1 of 8
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Economics 4261 Introduction to Econometrics Fall 2010 Question 2 Let A 1 2 1 4 3 0 3 1 0 . (a) A 1 A . (b) Calculate A ¡ 1 . (c) Solve A ¤ x ¤ ¥ 3 2 1 . .................................................................................................................. (a)
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Homework1Key - Economics 4261 Introduction to Econometrics...

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