516hw9 - Section 4.5 6 a 1 1/8 = 7/8 b 2 3 1 0 otherwise x...

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Unformatted text preview: Section 4.5 6. a) 1 - 1/8 = 7/8. b) 2 3 1 ( ) 0 otherwise x x f x f f = c) ( ) 3/ 4. E X = d) 3 1 2 3 1 2 3 ( ) ( , , ) ( ) ( ) ( ) independence P X x P Y x Y x Y x P Y x P Y x P Y x x = = = Section 4.6 1. Let i X denote the gap (in minutes) between the i th person’s arrival time and noon. Then i X ’s are i.i.d. N ( 0, 5 2 ). a) (1st person arrives before 11:50) 1 (no one arrives before 11:50) P P = - ( 29 1 2 3 4 4 1 ( 10, 10, 10, 10) 1 1 ( 2) 0.0881 P X X X X = ----- = -- Φ - = b) (Some of the 4 still not arrived at 12:15) 1 (all have arrived by 12:15) P P = - ( 29 4 1 (3) 0.0056 = - Φ = c) Let (2) X be the 2 nd order statistic of the i X ’s. Then the density of (2) X is given by ( 29 ( 29 ( 2) 2 2 3 ( ) 4 ( ) ( ) 1 ( ) 12 ( ) ( ) 1 ( ) 2 X f x f x F x F x f x F x F x =- =- Where f ( x ) and F ( x ) are the density and distribution functions of i X ’s. Over the interval [- 1/6, 1/6] this density could be considered constant. So (2) (2 person arrives within 10 seconds of noon) ( 1/ 6 1/ 6) P nd P X =- [Recall that our measurements were in minutes.] ( 29 (2) 2 ([ 1/ 6,1/ 6]) (0) 1 12 (0) (0) 1 (0) 3 0.0399 X length f f F F f- =- Section 4.4 3. Change of variable formula tells us that if 3....
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This note was uploaded on 09/14/2009 for the course STAT 134 taught by Professor Aldous during the Fall '03 term at Berkeley.

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516hw9 - Section 4.5 6 a 1 1/8 = 7/8 b 2 3 1 0 otherwise x...

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