Hw07ans - STAT 410 Spring 2008 Homework#7(due Friday March...

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Unformatted text preview: STAT 410 Spring 2008 Homework #7 (due Friday, March 7, by 3:00 p.m.) 1. Suppose X has a multivariate normal N 3 ( μ , Σ ) distribution with mean μ = & & & ¡ ¢ £ £ £ ¤ ¥ 23 17 7 and covariance matrix Σ = & & & ¡ ¢ £ £ £ ¤ ¥-- 25 10 10 9 2 2 4 . a) Find P ( X 1 > 10 ). X 1 ~ N ( 7, 4 ) P ( X 1 > 10 ) = & ¡ ¢ £ ¤ ¥- > 2 7 10 Z P = P ( Z > 1.50 ) = 0.0668 . b) Find P ( X 1 > 10 | X 2 = 20, X 3 = 25 ). 11 & = ( 4 ) 12 & = ( 2 0 ) & & ¡ ¢ £ £ ¤ ¥ = 2 21 & & & ¡ ¢ £ £ ¤ ¥-- = 25 10 10 9 22 & & & ¡ ¢ £ £ ¤ ¥ & & ¡ ¢ £ £ ¤ ¥ = =- 072 . 08 . 08 . 20 . 9 10 10 25 125 1 1 22 & ( ) ( ) 16 . 40 . 072 . 08 . 08 . 20 . 2 1 22 12 = = & & ¡ ¢ £ £ ¤ ¥- & & ( ) 2 2 1 22 12 1 ¡ X & & ¡- +- = 7 + ( ) & & ¡ ¢ £ £ ¤ ¥-- 23 25 17 20 16 . 40 . = 8.52. 21 1 22 12 11 & & & &-- = 4 – ( ) & & ¡ ¢ £ £ ¤ ¥ 2 16 . 40 . = 3.2. X 1 | X 2 = 20, X 3 = 25 ~ N ( 8.52, 3.2 ) P ( X 1 > 10 | X 2 = 20, X 3 = 25 ) = & & ¡ ¢ £ £ ¤ ¥- > 2 . 3 52 . 8 10 Z P = P ( Z > 0.83 ) = 0.2033 . c) Find P ( X 2 < 20 ). X 2 ~ N ( 17, 9 ) P ( X 2 < 20 ) = & ¡ ¢ £ ¤ ¥- < 3 17 20 Z P = P ( Z < 1.00 ) = 0.8413 . d) Find P ( X 2 < 20 | X 1 = 10, X 3 = 20 ). 11 & = ( 9 ) 12 & = ( 2 – 10 ) & & ¡ ¢ £ £ ¤ ¥- = 10 2 21 & & & ¡ ¢ £ £ ¤ ¥ = 25 4 22 & & & ¡ ¢ £ £ ¤ ¥ & & ¡ ¢ £ £ ¤ ¥ = =- 04 . 25 . 4 25 100 1 1 22 & ( ) ( ) 40 . 50 . 04 . 25 . 10 2 1 22 12- & & ¡ ¢ £ £ ¤ ¥- = =- & & ( ) 2 2 1 22 12 1 ¡ X & & ¡- +- = 17 + ( ) & & ¡ ¢ £ £ ¤ ¥--- 23 20 7 10 40 . 50 . = 19.7. 21 1 22 12 11 & & & &-- = 9 – ( ) & & ¡ ¢ £ £ ¤ ¥-- 10 2 40 . 50 . = 4. X 2 | X 1 = 10, X 3 = 20 ~ N ( 19.7, 4 ) P ( X 2 < 20 | X 1 = 10, X 3 = 20 ) = & ¡ ¢ £ ¤ ¥- < 2 7 . 19 20 Z P = P ( Z < 0.15 ) = 0.5596 ....
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Hw07ans - STAT 410 Spring 2008 Homework#7(due Friday March...

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