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sec3_f11 - Econ 140 Fall 2011 Economics 140 1 Homework#1...

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Econ 140 Fall 2011 Economics 140 September 13, 2011 1 Homework #1 Due September 21, 2011 at beginning of lecture Chapter 2: 4, 6, C7 More problems will be added before the due date. Please be sure to submit either a do- or log file of your Stata work with your problem set. Only hard copies will be accepted; no emailed assignments, please. 2 Some more math Some properties of expectations: Expectation of functions: E [ g ( Y )] = X y Y f ( y ) g ( y ) (discrete) E [ g ( Y )] = Z -∞ f ( y ) g ( y ) dy (continuous) Expectation of a function is not equal to the function of the expectation: E [ g ( y )] 6 = g ( E [ Y ]) Linearity: E [ ag ( Y ) + bh ( Y ) + c ] = aE [ g ( Y )] + bE [ h ( Y )] + E [ c ] = aE [ g ( Y )] + bE [ h ( Y )] + c E [ X + Y ] = E [ X ] + E [ Y ] E [ N X i =1 A i ] = N X i =1 E [ A i ] Law of Iterated Expectations : E [ Y ] = X X x =1 E [ Y | X ] Pr [ X = x ] (discrete) E [ Y ] = Z -∞ E [ Y | X ] f ( x ) (continuous) Suppose we are trying to calculate the expected value of weight (Y), and we have information about gender (X) as well. In this case, we can use the law of iterated expectations to use information about average weight, given gender, to estimate the average weight of a person: E [ Y ] = 2 X x =1 E [ Y | X ] Pr [ X = x ] = E [ weight | male ] Pr [ X = male ] + E [ weight | female ] Pr [ X = female ] 1
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Econ 140 Fall 2011 3 Some definitions Recall how we can characterize the distribution of a random variable with moments: 1. The expected value (mean) is the value we expect the random variable to take in the population, or the measure of central tendency.
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