# Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant . Let* = Annie’s arrival time and Y

Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant . Let* = Annie’s arrival time and Y = Alvie’s arrival time .Suppose X and Y are independent with each uniformly distributed on the interval [ 5 , 6].( a) What is the joint put of X and Y ?0flay ) = \ 2 4 5 Says 6otherwise*~ } ( 2 . 4 ) = | ( .2 – 5 ) . ( 4 – 5 ) 5 5 1 5 6: 5 5 4 5 0Ootherwise*\1 0 &lt; 2: 0 cy~ } ( * * 4) = } 0 otherwise*5 &lt; &lt; &lt; 6 : 5 5 4 5 6\f ( * * 4 ) = 9 0 otherwise*( b) What is the probability that they both arrive between 5: 12 and 5:39 ? ( Give answer accurate to 3 decimal places . )( C ) If the first one to arrive will wait only 15 min before leaving to eat elsewhere , what is the probability that they have dinner at the health-food restaurant ? [ Hint :&quot;The event of interest is A = ( ( XX) : 1 x – V 1 5 1/4 1 . I ( Give answer accurate to 3 decimal places . )

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