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**Unformatted text preview: **Eighth Homework Set — Solutions Chapter 5 Problem 15 (a) P { X > 5 } = P braceleftbig X- 10 6 > 5- 10 6 bracerightbig = 1- Φ (- 5 6 ) = Φ ( 5 6 ) = 0 . 7977 (b) P { 4 < X < 16 } = P braceleftbig- 1 < X- 10 6 < 1 bracerightbig = Φ (1)- Φ (- 1) = 2Φ (1)- 1 = 0 . 6827 (c) P { X < 8 } = P braceleftbig X- 10 6 <- 1 3 bracerightbig = Φ (- 1 3 ) = 1- Φ ( 1 3 ) = 0 . 3695 (d) P { X < 20 } = P braceleftbig X- 10 6 <- 10 6 bracerightbig = Φ ( 5 3 ) = 0 . 9522 (e) P { X > 16 } = P braceleftbig X- 10 6 > 1 bracerightbig = 1- Φ (1) = 0 . 1587 Problem 18 We have P { X > 9 } = P braceleftbig X- 5 σ > 4 σ bracerightbig = 1- Φ ( 4 σ ) = 0 . 2, so that Φ ( 4 σ ) = 0 . 8, hence 4 σ = 0 . 85. This implies that σ = 4 . 7059, so that the variance is σ 2 = 22 . 145. Problem 21 Let X be a normal random variable with μ = 71 and σ 2 = 6 . 25. Then P { X > 74 } = P braceleftbig X- 71 2 . 5 > 3 2 . 5 bracerightbig = 1- Φ ( 6 5 ) = 0 . 1151....

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