HW01-AK - HOMEWORK 1 ANSWER KEY, February 14, 2011 4751,...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 10/05/2011 for the course ECON 4751 taught by Professor Staff during the Summer '08 term at Minnesota.

Page1 / 5

HW01-AK - HOMEWORK 1 ANSWER KEY, February 14, 2011 4751,...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online