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Unformatted text preview: Final Practice Problems  Answer Key 1. The following is a fictional joint probability distribution for level of household income and the type of computer a household owns. Computer type is entered in the left most column, and level of income on the top most row. Computer Type/Income 15,000 40,000 90,000 No Computer 0.2 0.1 PC 0.1 0.2 0.1 Mac 0.1 0.2 (a) What is the probability that household income is 15,000? What is the probability household income is 40,000? What is the probability that household income is 90,000? Let I denote income, C type of computer NC no computer and M denote Mac. We then obtain: P ( I = 15 , 000) = P ( I = 15 , 000; C = NC )+ P ( I = 40 , 000; C = PC )+ P ( I = 90 , 000; C = M ) = 0 . 3 Similarly, we obtain P ( I = 40 , 000) = 0 . 4 and P ( I = 90 , 000) = 0 . 3. (b) What is the probability that household income is higher than 30,000? Since P ( I 30 , 000) = P ( I = 40 , 000) + P ( I = 90 , 000) from (a) we obtain: P ( I 30 , 000) = 0 . 4 + 0 . 3 = 0 . 7 (c) Conditional on income being 90,000, what is the probability that a household owns a PC? Using our answer from (a) we quickly obtain: P ( C = PC  I = 90 , 000) = P ( C = PC ; I = 90 , 000) P ( I = 90 , 000) = 1 3 (d) Conditional on income being 90,000, what is the probability that a household owns a Mac? Following (d) we obtain P ( C = M  I = 90 , 000) = 2 / 3. (e) Are the level of income and computer ownership independent from each other? They are not. It is easy to verify, for example, by noting that: P ( I = 90 , 000; C = NC ) = 0 6 = 0 . 3 . 3 = P ( I = 90 , 000) P ( C = NC ) 2. Suppose X is normally distributed with unknown mean and variance 2 . You sample 25 observa tions and find X = 5 and s 2 X = 16....
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 Spring '10
 Elliot

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