Quartile deviation is 1−1 =0.2063 (solving F (x)=x3

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Quartile deviation is 1−1 =0.2063 (solving F (x)=x3 Powered By Docstoc
					2
                                                   √      R 1.02      ³       2      2
                                                                                       ´
                     e
    1. (a) Pr(0.99 < X < 1.02) =                 3· 218
                                                  √                exp − 218·3 ·(x−1) dx = 48.33%
                                                    2π     0.99                 2
                                                                                       3
             Quartile deviation is 1 − 21/3 = 0.2063 (solving F (x) = x = 1 for
                                                   1
                                                                      2   4
             x yields lower quartile).
                                          √ √     R        ³            ´
         (b) Pr(0.99 < X ¯ < 1.02) = 10· 218 1.02 exp − 218·10·(x−1)2 dx =
                                            √
                                             2π     0.99           2
                                          1
               50.45% (since σ 2 =       10
                                            ).

    2.
                                          ¡¢          ¡¢
         (a) p = 1 − 2.8e−1.8 = 0. 537163, 4 p2 q2 + 4 p3 q + p4 = 74.11% since
                                           2           3
             F (x) = 1 − (1 + x)e−x
                R
             4! ∞
         (b) 21 0 x (1 − (1 + x)e−x ) (1 + x)2 e−2x xe−x dx = 1.438
                                 √      R 0.29      ³       2         2
                                                                        ´
          e
    3. Pr(X < 0.29) =          2· 301
                                √                exp − 301·2 ·(x−0.25) dx = 91.74% based
                                  2π     −∞                    2
                 √
       on F (x) = x
                                   9/16−1/16
         Quartile deviation is         2
                                                  = 1.
                                                    4

    4.
                      √       ¡¢         ¡¢         ¡¢
          (a) p =      .6, 1 − 8 p5 q 3 − 8 p6 q 2 − 8 p7 q − p8 = 8.2585%
                               5          6          7

                8!
                      R √ 4
                       1        √                    R1       √ 3   √
         (b)   4!3!
                                      1
                         x x (1− x)3 2√x dx = 1 , 140 (x− 1 )2 x (1− x)3 dx =
                                              3           3
                                                                                             1
                                                                                             33
                      0                                            0


    5. f (x, y) = n(n−1)(y−x)n−2 for 0 < x < y < 1 ⇒ f (x, r) = n(n−1)rn−2
       for 0 < x < 1 and 0 < r < 1 − x (where r = y − x) ⇒ f (r) =
                      R
                     1−r
       n(n − 1)rn−2      dx = n(n − 1)rn−2 (1 − r) for 0 < r < 1, which is
                           0
         beta(n − 1, 2).
                                        R1                               R1
                                    7!                                7!
         When n = 7, E[X(1) ] = 0!×6! 0 x(1−x)6 dx = 1 , E[X(7) ] = 6!×0! 0 x7 dx =
                                                     8
         7
                                         R1 Ry
         8
           and E[X(1) · X(7) ] = 7 × 6 0 y 0 x(y − x)5 dxdy = 1 , which implies
                                                                 9
                                       7
         that Cov[X(1) , X(7) ] = 1 − 64 = 0.001736
                                  9
                                                        R 1 R 0.1
         Finally, Pr[X(1) < 0.1 ∩ X(7) > 0.9] = 7 × 6 0.9 0 (y − x)5 dxdy =
         25.31%

				
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posted:8/31/2010
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