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hw5sol - ORIE 3500/5500 Fall Term 2008 Assignment...

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ORIE 3500/5500 Fall Term 2008 Assignment 5-Solution 1. Let J and F be the number of days with snowfall in January and February. Then J Binomial (31 , 0 . 4) and F Binomial (28 , 0 . 4). (a) P ( J = 4) = ( 31 4 ) 0 . 4 4 0 . 6 27 . i. Since there are 4 Sundays in January, the probability that in all of them there will be snowfall is ( 4 4 ) 0 . 4 4 0 . 6 27 . Hence the required probability is ( 4 4 ) 0 . 4 4 0 . 6 27 ( 31 4 ) 0 . 4 4 0 . 6 27 = 1 ( 31 4 ) . ii. Since there are 8 days which are either Sundays or Saturdays, u sing similar argument as above the required probability is ( 8 4 ) 0 . 4 4 0 . 6 27 ( 31 4 ) 0 . 4 4 0 . 6 27 = ( 8 4 ) ( 31 4 ) . (b) For r = 0 , 1 , . . . , 59, P ( J + F = r ) = r i =0 P ( J = i, F = r - i ) = r i =0 P ( J = i ) P ( F = r - i ) = r i =0 31 i 0 . 4 i 0 . 6 31 - i 28 r - i 0 . 4 r - i 0 . 6 28 - r + i = 0 . 4 r 0 . 6 59 - r r i =0 31 i 28 r - i = 59 r 0 . 4 r 0 . 6 59 - r . 1
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(c) For x = 9 , 10 , . . . , 28, we have P ( F = x | J + F = 40) = P ( F = x, J = 40 - x ) P ( J + F = 40) = P ( F = x ) P ( J = 40 - x ) P ( J + F = 40) = ( 28 x ) 0 . 4 x 0 . 6 28 - x ( 31 40 - x ) 0 . 4 40 - x 0 . 6 x - 9 ( 59 40 ) 0 . 4 40 0 . 6 19 = ( 28 x )( 31 40 - x ) ( 59 40 ) . 2. For o x 1, P ( U 1 x ) = P ( U 2 x ) = x . (a) For 0 v w 1, F ( v, w ) = P ( V v, W w ) = P ( U 1 , U 2 w ) - P ( v < U 1 , U 2 w ) = P ( U 1 w ) P ( U 2 w ) - P ( v < U 1 w ) P ( v < U 2 w ) = w 2 - ( w - v ) 2 = 2 vw - v 2 .
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