410Hw03ans - STAT 410 Summer 2009 Homework#3(due Monday...

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STAT 410 Summer 2009 Homework #3 (due Monday, June 29, by 4:00 p.m.) 1. Suppose Heidi has a fair 4-sided die, and Alex has a fair 6-sided die. Each day, they roll their dice (independently) until someone rolls a “1”. (Then the person who did not roll a “1” does the dishes.) Find the probability that … p H ( x ) = - 4 1 4 3 1 x , x = 1, 2, 3, … , p A ( y ) = - 6 1 6 5 1 y , y = 1, 2, 3, … . a) they roll the first “1” at the same time (after equal number of attempts); ( 29 ( 29 = 1 A H k k p k p = = - - 1 1 1 6 1 6 5 4 1 4 3 k k k = 9 1 . b) it takes Alex twice as many attempts as it does Heidi to roll the first “1”; ( 29 ( 29 = 1 A H 2 k k p k p = = - - 1 1 2 1 6 1 6 5 4 1 4 3 k k k = 69 5 . c) Alex rolls the first “1” before Heidi does; ( 29 ( 29 = + = 1 1 A H k k m k p m p = = + = - - 1 1 1 1 4 1 4 3 6 1 6 5 k k m m k = 3 1 . d) Alex rolls the first “1” on an even-numbered attempt; ( 29 = 1 A 2 k k p = = - 1 1 2 6 1 6 5 k k = 11 5 . e) Alex rolls a “1” before he rolls an even number. = - 1 1 6 1 6 2 k k = 4 1 . [ 2 / 6 = P( not “1” and not even ) ]
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2. Let X, Y, and Z be i.i.d. Uniform [ 0 , 1 ] random variables Find the probability distribution of W = X + Y + Z. That is, find ( 29 w f W . Hint: If V = X + Y, we know the p.d.f. of V, f V ( v ) ( see Examples for 06/24/2009 ): f V ( v ) = v if 0 < v < 1, f V ( v ) = 2 – v if 1 < v < 2, f V ( v ) = 0 otherwise. Now use convolution formula to find the p.d.f. of W = V + Z. There will be 5 possible cases; two of them are “boring”, two of them are “exciting”, and one is “really exciting”. ( 29 < < = otherwise 0 1 0 1 Z z z f ( 29 < < - = < - < = - otherwise 0 1 1 otherwise 0 1 0 1 Z w v w v w v w f ( 29 w f W = ( 29 ( 29 ( 29 - = - + dv v w f v f w f Z V Z V (convolution) Case 1 : w < 0. ( 29 w f W = 0.
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Case 2 : 0 < w < 1. Then w – 1 < 0. ( 29 ( 29 2 1 2 0 W w dv v w f w = = . Case 3 : 1 < w < 2. Then 0 < w – 1 < 1. ( 29 ( 29 ( 29 ( 29 2 3 3 1 2 1 2 1 1 1 W - + - = - + = - w w dv v dv v w f w w ( 29 ( 29 2 1 2 1 + - - = w w .
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410Hw03ans - STAT 410 Summer 2009 Homework#3(due Monday...

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