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ps1_ak - Problem Set 1 Introduction to Econometrics Fall...

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Problem Set 1 Introduction to Econometrics - Fall 2006 Due Date: Thursday, September 21st, BEFORE CLASS. Each student is responsible for handing one handwritten solution. If working in groups, indicate the members of the group (to facilitate grading). PLEASE SHOW YOUR WORK . 1. The random variable X is distributed normally with N ° ° X ; ± 2 X ± (a) What is the Pr( X ° ° X ) ? What is the Pr( X ° ° X + 1 : 96 ± X ) ? Solution: 0 : 5 , 0 : 975 (b) Let Y = X ° ° X ± X . Show that Y is distributed normally with N (0 ; 1) ; that is E ( Y ) = 0 and V ar ( Y ) = 1 . Solution: Just use formulas at page 26: E [ Y ] = E ² X ± ° X ± X ³ = 1 ± X E [ X ± ° X ] since 1 ± X is a constant. Moreover: E [ Y ] = 1 ± X E [ X ± ° X ] = 1 ± X ( E [ X ] ± ° X ) since ° X is another constant. We know E [ X ] = ° X ; then: E [ Y ] = 1 ± X ( E [ X ] ± ° X ) = 1 ± X 0 Consequently: E [ Y ] = 0 Next, regarding V ar ( Y ) ; by de°nition: V ar ( Y ) = E h ( Y ± E [ Y ]) 2 i = E " ´ X ± ° X ± X ± 0 µ 2 # = E " ´ X ± ° X ± X µ 2 # = E ² 1 ± 2 X ( X ± ° X ) 2 ³ = 1 ± 2 X E h ( X ± ° X ) 2 i since 1 ± 2 X is a constant. We know that ± 2 X = E h ( X ± ° X ) 2 i by de°nition, so: V ar ( Y ) = 1 ± 2 X E h ( X ± ° X ) 2 i = ± 2 X ± 2 X = 1 1
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(c) Show that E ° Y 2 ± = 1 Solution: either use the de°nition of Y = X ° ° X ± X and compute E ´ X ° ° X ± X · 2 µ , or use the fact that V ar ( Y ) = E ° Y 2 ± ± [ E ( Y )] 2 By de°nition: E ° Y 2 ± = E " ´ X ± ° X ± X µ 2 # = E ² 1 ± 2 X ( X ± ° X ) 2 ³ = 1 ± 2 X E h ( X ± ° X ) 2 i since 1 ± 2 X is a constant. We know that ± 2 X = E h ( X ± ° X ) 2 i by de°nition, so: E ° Y 2 ± = 1 ± 2 X E h ( X ± ° X ) 2 i = ± 2 X ± 2 X = 1 On the other side, we could have noticed that V ar ( Y ) = E ° Y 2 ± ± [ E ( Y )] 2 In this case, E ( Y ) = 0 ; so [ E ( Y )] 2 = 0 : Then: E ° Y 2 ± = V ar ( Y ) = 1 2. Let the random variable X be the temperature in Madison, WI at 12:00pm on September 21th. You know that X is distributed normally with N (72 ; 100) . (a) What is the probability the temperature is lower than 46 ± F ? What is the probability the temperature is higher than 90 ± F ? What is the probability the temperature is in between 65 ± F and 80 ± F ? Higher than 80 ± F ?
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