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Unformatted text preview: HOMEWORK 1 ANSWER KEY, February 14, 2011 4751, Spring 2011 February 14, 2011 Due date: In class February 14th. • You are free to work with your colleagues, in fact I encourage you to do so. However, each of you must submit your own answer and indicate with whom you have worked with. • Homeworks must be typed, read the CLA policy (or our syllabus) concerning this. • When turning in your answers, write your responses in numbered order (there is no need to submit a printed version of this document). The first exercise is to compute Mean and Variance. You can use Excel to do this, I encourage you to do it by hand first. Exercise 1. Let R be a random variable (r.v.) describing ouput for some firm. Assume R is: with probability 1 / 4 , 1 with probability 1 / 2 , 2 with probability 1 / 8 and 3 with probability 1 / 8 . The profit is a function of R , we will denote it by P given by the square of output. 1 (a) Write down the probability distribution of P as a table going from the smallest value of P to the largest. Answer. P = R 2 , takes the values , 1 , 4 , 9 with respective probability distribution 1 4 , 1 2 , 1 8 , 1 8 , (i.e. the distribution is given by p = (( 0; 1 4 ) , ( 1; 1 2 ) , ( 4; 1 8 ) , ( 9; 1 8 )) . (b) Compute the expected value of R and of P (denoted E ( R ) and E ( P ) ). ER = 0 1 4 +1 1 2 +2 1 8 +3 1 8 = 0+4+2+3 8 = 9 8 = 1 . 125 , EP = 0 1 4 +1 1 2 +4 1 8 +9 1 8 = 0+4+4+9 8 = 17 8 = 2 . 125 (c) Compute the variance of R and of P (denoted V AR ( R ) and V AR ( P ) or σ 2 R and σ 2 P ). Answer. In this question it would be useful to use this equality to compute the variance σ 2 R := ∑ s p s ( R ( s ) ER ) 2 = E ( R 2 ) ( ER ) 2 , 2 Hence, σ 2 R = E ( P ) ( ER ) 2 = ( 17 8 ) ( 9 8 ) 2 = 0 . 859375 and σ 2 P = E ( P 2 ) ( 17 8 ) 2 = 8 . 109375 , where E ( P 2 ) = 0 1 4 + 1 1 2 + 16 1 8 + 81 1 8 = 101 8 (d) Compute the standard deviation of R and of P (denoted σ R and σ P ). Answer. σ R = p Var ( R ) = 0 . 927024811 and σ P = 2 . 847696437 (e) Compute the covariance between R and P (denoted COV ( R,P ) ). Answer. Here is the algebra 1 thanks goes to Joshua miller who came up with this question 2 σ 2 R = ∑ p s R ( s ) 2 2 R ( s ) ER + ( ER ) 2 = ∑ p s R ( s ) 2 2 ER ∑ s p s R ( s ) + ∑ s p s ( ER ) 2 = ∑ p s R ( s ) 2 2 ER ∑ s p s R ( s ) + ∑ s p s ( ER ) 2 = E ( R 2 ) 2 ER · ER + ( ER ) 2 · 1 = E ( R 2 ) ( ER ) 2 1 4751, Spring 2011 February 14, 2011 Rene Schwenger Cov ( R,P ) = E (( R E ( R )) · ( P E ( P ))) = X Prob ( R = r,P = p ) ( R μ R ) ( P μ P ) = 1 4 (0 μ R ) (0 μ P ) + 1 2 (1 μ R ) (1 μ P ) + 1 8 (2 μ R ) (4 μ P ) + 1 8 (3 μ R ) (9 μ P ) = 2 . 484375 The interested reader may find the following discusion useful....
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This note was uploaded on 10/05/2011 for the course ECON 4751 taught by Professor Staff during the Summer '08 term at Minnesota.
 Summer '08
 Staff

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