120M25A - Midterm Exam 2 Practice Problems Answer Key 1...

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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 0 10 years 0 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 × 0 + 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. (d) Are years of experience and salaries independent? Justify your answer. Since the covariance is positive we know that they are not independent. (e) The current exchange rate between US dollar and Euro is 1 Dollar = 0.65 Euro. What is the covariance between years of experience and salaries in Euros. That is, what would the covariance 1
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