120M25A - 1 Consider the following joint probability...

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Unformatted text preview: 1. Consider the following joint probability distribution of years of experience in a job and annual salary. Experience / Salary $30 , 000 $60 , 000 $90 , 000 0 years 1/3 1/6 10 years 1/6 1/3 (a) What is the probability of having 0 years of experience? What is the probability of having 10 years of experience? Let E be years of experience and S be salary. To find P ( E = 0) use the formula for a marginal: P ( E = 0) = P ( E = 0 ,S = 30 , 000)+ P ( E = 0 ,S = 50 , 000)+ P ( E = 0 ,S = 90 , 000) = 1 3 + 1 6 +0 = 1 2 We can find P ( E = 10) in the same manner, or by noticing that P ( E = 0) + P ( E = 10) = 1, which implies that P ( E = 10) = 1- P ( E = 0) = 1 / 2. (b) What is the probability of earning $30 , 000 conditional on having 0 years of experience? What is the probability of earning $30 , 000 conditional on having 10 years of experience? For this question, use the formula for conditional probabilities and the answers to part (a) to find that P ( S = 30 , 000 | E = 0) = P ( S = 30 , 000 ,E = 0) /P ( E = 0) = (1 / 3) / (1 / 2) = 2 / 3. Similarly, P ( S = 30 , 000 | E = 10) = P ( S = 30 , 000 ,E = 10) /P ( E = 10) = 0 / (1 / 2) = 0. (c) What is the covariance between years of experience and salary earned? To answer this question we use the formula Cov ( E,S ) = E [ ES ]- E [ E ] × E [ S ]. To find E [ ES ]: E [ ES ] = 0 × 30 , 000 × 1 3 + 0 × 60 , 000 × 1 6 + 0 × 90 , 000 × + 10 × 30 , 000 × 0 + 10 × 60 , 000 × 1 6 + 10 × 90 , 000 × 1 3 = 400 , 000 Using our answer to (a), we can find E [ E ] = 0 × P ( E = 0) + 10 × P ( E = 10) = 5, while in a similar way we obtain E [ S ] by noting: E [ S ] = 30 , 000 × P ( S = 30 , 000) + 60 , 000 × P ( S = 60 , 000) + 90 , 000 × P ( S = 90 , 000) 30 , 000 3 + 60 , 000 3 + 90 , 000 3 = 60 , 000 We can then find E [ ES ]- E [ E ] × E [ S ] = 400 , 000- 5 × 60 , 000 = 100 , 000....
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This note was uploaded on 12/05/2010 for the course ECON 120A taught by Professor M.abajian during the Spring '10 term at San Diego.

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120M25A - 1 Consider the following joint probability...

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