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398_39_solutions-instructor-manual_Review_Chapter.pdf

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Department
Economics
Course
Economics 2122A/B
Professor
Terry Biggs
Semester
Winter

Description
Dougherty Introduction to Econometrics 4e Instructors Manual REVIEW RANDOM VARIABLES SAMPLING AND ESTIMATIONOften the problems experienced by students with econometrics have nothing to do with econometrics as such but arise as a consequence of gaps in the understanding of the statistical infrastructureThe function of the present chapter is not to attempt to serve as a substitute for a statistics course which should be a nonnegotiable prerequisiteIts aim is to provide an opportunity to ensure that students are really up to speed on concepts such as unbiasedness efficiency and consistency particularly the latter since asymptotics are often neglected in standard statistics courses R1 IntroductionR2 Discrete random variables and expectationsR1 A random variable X is defined to be the difference between the higher value and the lower value when two dice are thrownIf they have the same value X is defined to be zeroFind the probability distribution for X Answer The tables show the 36 possible outcomes and the corresponding probability distribution 1 2 3 4 5 6 redgreen1 0 1 2 3 4 52 1 0 1 2 3 43 2 1 0 1 2 34 3 2 1 0 1 25 4 3 2 1 0 16 5 4 3 2 1 0 Value of X 0 1 2 3 4 5 Frequency 6 10 8 6 4 2 Probability 636 1036 836 636 436 236R2 A random variable X is defined to be the larger of the two values when two dice are thrown or the value if the values are the sameFind the probability distribution for XNoteAnswers to exercises marked with an asterisk are provided in the Study Guide Answer The table shows the 36 possible outcomesThe probability distribution is derived by counting the number of times each outcome occurs and dividing by 36The probabilities have been written as fractions but they could equally well have been written as decimalsC Dougherty 2011 All rights reserved REVIEW RANDOM VARIABLES SAMPLING AND ESTIMATION 2 1 2 3 4 5 6 redgreen1 1 2 3 4 5 62 2 2 3 4 5 63 3 3 3 4 5 64 4 4 4 4 5 65 5 5 5 5 5 66 6 6 6 6 6 6Value of X 1 2 3 4 5 6 Frequency 1 3 5 7 9 11 Probability 136 336 536 736 936 1136R3 Find the expected value of X in Exercise R1 Answer The table is based on Table R2 in the textStudents should be told that is a good idea to guess the outcome before doing the arithmeticIn this case since the lower numbers have the largest probabilities the expected value should clearly be no more than 2If the calculated value does not conform with the guess it is possible that this is because the guess was poorHowever it may be because there is an error in the arithmetic and this is one way of catching such errorsX p Xp 06360 110361036 28361636 36361836 44361636 52361036 Total703619444R4 Find the expected value of X in Exercise R2 Answer It is a good idea to guess the outcome before doing the arithmeticIn this case since the higher numbers have the largest probabilities the expected value should clearly lie between 4 and 5If the calculated value does not conform with the guess it is possible that this is because the guess was poorHowever it may be because there is an error in the arithmetic and this is one way of catching such errorsC Dougherty 2011 All rights reserved
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