HW28Answers(10.3) - given that Cov ( X, Y ) = 0, X + Y = r...

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Math331, Spring 2008 Instructor: David Anderson Section 10.3 Homework Answers Homework: pgs. 433 - 434, #’s 1, 5, 6. 1. Cov ( X, Y ) = ρ ( X, Y ) σ X σ Y = (1 / 2) * 2 * 3 = 3 . Therefore, after Frst splitting o± the “RV” 3, then splitting up the X and Y terms we have V ar (2 X - 4 Y + 3) = V ar (2 X - 4 Y ) = 2 2 V ar ( X ) + 4 2 V ar ( Y ) - 2 * 2 * 4 Cov ( X, Y ) = 4 σ 2 X + 16 σ 2 Y - 16 * 3 = 4 * 4 + 16 * 9 - 48 = 112 . 5. No, because | ρ ( X, Y ) | ≤ 1 for any RVs X and Y . 6. We suppose that Cov ( X, Y ) = 0. By deFnition we have that, ρ ( X + Y, X - Y ) = Cov ( X + Y, X - Y ) σ X + Y σ X - Y Now, Cov ( X + Y, X - Y ) = E [( X + Y - ( E [ X ] + E [ Y ]))( X - Y - ( E [ X ] - E [ Y ]))] = E [(( X - E [ X ]) + ( Y - E [ Y ]))(( X - E [ X ]) - ( Y - E [ Y ]))] = E b ( X - E [ X ]) 2 - ( Y - E [ Y ]) 2 B = V ar ( X ) - V ar ( Y ) . You could also get this from quoting the result of # 8 of Section 10.2. Next, because we are
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Unformatted text preview: given that Cov ( X, Y ) = 0, X + Y = r V ar ( X + Y ) = r V ar ( X ) + V ar ( Y ) + 2 Cov ( X, Y ) = r V ar ( X ) + V ar ( Y ) X-Y = r V ar ( X-Y ) = r V ar ( X ) + V ar ( Y )-2 Cov ( X, Y ) = r V ar ( X ) + V ar ( Y ) = X + Y X-Y = V ar ( X ) + V ar ( Y ) . Putting all of the above together gives us ( X + Y, X-Y ) = Cov ( X + Y, X-Y ) X + Y X-Y = V ar ( X )-V ar ( Y ) V ar ( X ) + V ar ( Y ) . 1...
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This note was uploaded on 06/24/2008 for the course MATH 331 taught by Professor Anderson during the Spring '08 term at University of Wisconsin.

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