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Unformatted text preview: from which we obtain E [ X  A ] = Z 3 2 x Â· 2 x 5 dx = 2 x 3 15 âŽ© âŽ© âŽ© 3 2 = 54 15 âˆ’ 16 15 = 38 15 . (b) We have E [ Y ] = E [ X 2 ] = Z 3 1 x 3 4 dx = 5 , and E [ Y 2 ] = E [ X 4 ] = Z 3 1 x 5 4 dx = 91 3 . Thus, var( Y ) = E [ Y 2 ] âˆ’ ( E [ Y ] ) 2 = 91 3 âˆ’ 5 2 = 16 3 . Solution to Problem 3.19. (a) We have, using the normalization property, Z 2 1 cx âˆ’ 2 dx = 1 , or c = 1 Z 2 1 x âˆ’ 2 dx = 2 . (b) We have P ( A ) = Z 2 1 . 5 2 x âˆ’ 2 dx = 1 3 , and f X  A ( x  A ) = 6 x âˆ’ 2 , if 1 . 5 < x â‰¤ 2, , otherwise. (c) We have E [ Y  A ] = E [ X 2  A ] = Z 2 1 . 5 6 x âˆ’ 2 x 2 dx = 3 , E [ Y 2  A ] = E [ X 4  A ] = Z 2 1 . 5 6 x âˆ’ 2 x 4 dx = 37 4 , and var( Y  A ) = 37 4 âˆ’ 3 2 = 1 4 . Solution to Problem 3.20. The expected value in question is E [Time] = ( 5 + E [stay of 2nd student] ) Â· P (1st stays no more than 5 minutes) + ( E [stay of 1st  stay of 1st â‰¥ 5] + E [stay of 2nd] ) Â· P (1st stays more than 5 minutes) . 41 We have E [stay of 2nd student] = 30, and, using the memorylessness property of the exponential distribution, E [stay of 1st  stay of 1st â‰¥ 5] = 5 + E [stay of 1st] = 35 . Also P (1st student stays no more than 5 minutes) = 1 âˆ’ e âˆ’ 5 / 30 , P (1st student stays more than 5 minutes) = e âˆ’ 5 / 30 . By substitution we obtain E [Time] = (5 + 30) Â· (1 âˆ’ e âˆ’ 5 / 30 ) + (35 + 30) Â· e âˆ’ 5 / 30 = 35 + 30 Â· e âˆ’ 5 / 30 = 60 . 394 . Solution to Problem 3.21. (a) We have f Y ( y ) = 1 /l , for 0 â‰¤ y â‰¤ l . Furthermore, given the value y of Y , the random variable X is uniform in the interval [0 , y ]. Therefore, f X  Y ( x  y ) = 1 /y , for 0 â‰¤ x â‰¤ y . We conclude that f X,Y ( x, y ) = f Y ( y ) f X  Y ( x  y ) = ( 1 l Â· 1 y , â‰¤ x â‰¤ y â‰¤ l , , otherwise. (b) We have f X ( x ) = Z f X,Y ( x, y ) dy = Z l x 1 ly dy = 1 l ln( l/x ) , â‰¤ x â‰¤ l. (c) We have E [ X ] = Z l xf X ( x ) dx = Z l x l ln( l/x ) dx = l 4 . (d) The fraction Y /l of the stick that is left after the first break, and the further fraction X/Y of the stick that is left after the second break are independent. Furthermore, the random variables Y and X/Y are uniformly distributed over the sets [0 , l ] and [0 , 1], respectively, so that E [ Y ] = l/ 2 and E [ X/Y ] = 1 / 2. Thus, E [ X ] = E [ Y ] E h X Y i = l 2 Â· 1 2 = l 4 . Solution to Problem 3.22. Define coordinates such that the stick extends from position 0 (the left end) to position 1 (the right end). Denote the position of the first break by X and the position of the second break by Y . With method (ii), we have X < Y . With methods (i) and (iii), we assume that X < Y and we later account for the case Y < X by using symmetry. Under the assumption X < Y , the three pieces have lengths X , Y âˆ’ X , and 1 âˆ’ Y . In order that they form a triangle, the sum of the lengths of any two pieces must exceed the length of the third piece. Thus they form a triangle if X < ( Y âˆ’ X ) + (1 âˆ’ Y ) , ( Y âˆ’ X ) < X + (1 âˆ’ Y ) , (1 âˆ’ Y ) < X + ( Y âˆ’ X ) ....
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This note was uploaded on 01/11/2011 for the course MATH 170 taught by Professor Staff during the Spring '08 term at UCLA.
 Spring '08
 staff
 Math

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