Quiz2-2007-2- test 2- 610 question 1

Quiz2-2007-2 test 2 610 question 1

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Unformatted text preview: hat the test score of such a person is (a) (4%) above 125; 0.04779. Sol) Denote the score of the test as X . Since X ∼ (100, 152 ), we have P (X > 125) = 1 − Φ (b) (4%) between 90 and 110. 125 − 100 15 = 1 − Φ(1.667) = 1 − 0.95221 = 0.04779 0.4952. Sol) P (90 < X < 110) = 90 − 100 110 − 100 −Φ 15 15 Φ(0.667) − Φ(−0.667) = 2Φ(0.667) − 1 = 2 × 0.7476 − 1 = 0.4952 = Φ 4 9. (8%) If X1 and X2 are independent exponential random variables each having parameter λ, find the joint density function of Y1 = X1 + X2 and Y2 = eX1 . fY1 ,Y2 (y1 , y2 ) = λ2 −λy1 , y2 e 0 < ln y2 < y1 < ∞. Sol) The joint density function of X1 and X2 is fX1 ,X2 (x1 , x2 ) = λ2 e−λ(x1 +x2 ) , x1 , x2 > 0. fY1 ,Y2 (y1 , y2 ) can be found through the following steps: • Expressing X1 and X2 in terms of Y1 and Y2 , then X1 ln Y2 X2 ...
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This document was uploaded on 11/01/2013.

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