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Solution Manual for Mathematical Statistics with Applications, 8th Edition

Struggling with textbook problems? Solution Manual for Mathematical Statistics with Applications, 8th Edition offers a clear breakdown of every exercise for easy understanding.

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    S OLUTIONS M ANUAL J OHN E. F REUND ’ S M ATHEMATICAL S TATISTICS WITH A PPLICATIONS E IGHTH E DITION Irwin Miller Maryless Miller

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    iii Table of Contents Chapter 1 .............................................................................................................................. 1 Chapter 2 .............................................................................................................................. 8 Chapter 3 ............................................................................................................................ 23 Chapter 4 ............................................................................................................................. 46 Chapter 5 ............................................................................................................................. 61 Chapter 6 ............................................................................................................................. 80 Chapter 7 ............................................................................................................................ 95 Chapter 8 .......................................................................................................................... 112 Chapter 9 ........................................................................................................................... 130 Chapter 10 ....................................................................................................................... 139 Chapter 11 ....................................................................................................................... 162 Chapter 12 ....................................................................................................................... 171 Chapter 13 ....................................................................................................................... 182 Chapter 14 ....................................................................................................................... 199 Chapter 15 ....................................................................................................................... 226 Chapter 16 ....................................................................................................................... 245 Appendix A ...................................................................................................................... 258

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    1 Chapter 1 1.1 (a) 1 2 1 n i i n   (b) 0 1 2 3 0 1 2 3 0 1 2 0 1 13   1.2 1 1 2 2 1 2 1 1 n n i i n i n n n       1.3 (a) 300 4 n  320 3 n  301 4 n  321 2 n  302 3 n  322 1 n  303 2 n  330 2 n  310 4 n  331 1 n  311 3 n  312 2 n  313 1 n  (b) 4 4 3 ... 2 1 32         1.4 1 2 2 1 2 3 1 1 n n i j n n n n    

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    2 Mathematical Statistics, 8E 1.5 (b) 6, 20, and 70 “2 out of 3” m = 2 1 2 2 2(1 2) 6 1 1                       “3 out of 5” m = 3 2 3 4 2 2(1 3 6) 20 2 2 2                               “4 out of 7” m = 4 3 4 5 6 2 2(1 4 10 20) 70 3 3 3 3                                       1.6 (a) 10 10 10 10! 0 (7.92665)(3.678797) (7.92665)(454,002.49) 3,598,719 e π            % error 3.6288 3.5987 100 0.83% 3.6288     12 12 12 12! 24 (8.683215)(4.41455) 475,683,224 e π          4.7800 4.7568 % error 100 0.69% 4.7900     (b) 52 13 39 52 52 52 13 39 39 39 52 104 52 52! 13 13! 39! 13 39 26 78 13 4 4 639 billion 19.5 13 13 3 19.5 3 e e e π π π π π                                 1.7 Using Stirling’s formula in 2 2 ! ! ! n n n n n        yields 2 2 2 2 2 2 4 1 2 2 2 n n n n n n n n n n e n e π π π π π                              1.8 r n and 3 12 = 1,728 1.9 1 5 3 1 7 and 21 5 5 r n r                         1.10 Substitute r n  for r into result of 1.9 1 1 5 1 4 and 6 5 3 2 r n n r r n r n                                   

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    Chapter 1 3 1.11 (b) Seventh row is 1, 6, 15, 20, 15, 6, 1 Eighth row is 1, 7, 21, 35, 35, 21, 7, 1 6 6 5 4 2 2 3 2 4 5 6 ( ) 6 15 20 15 6 x y x x y x y x y x y xy y         7 7 6 5 2 4 3 3 4 2 5 6 7 ( ) 7 21 35 35 21 7 x y x x y x y x y x y x y xy y          1.14 (a) Set x = 1 and y = 1 (b) Set x = 1 and y =  1 (c) Set x = 1 and y = a  1 1.19 (a) 1 1 3 5 15 2 2 2 2 24 384                        and ( 3)( 4)( 5) 10 6      (b) 1 2 1 5 2 1 4         2 3 1 1 1 1 1 1 1 3 1 2 1 2 4 2 2 4 2 2 2 4 1 1 3 512 64 8 3 2 1 2 8 64 512 512 571 2 2.23 512                                                                       1142 2.230 512  1.20 (a) ( 1)( 2)...( ) ( 1) ! r r r      (b) ( )( 1)...( 1) ( 1)...( 1) ( 1) ! ! 1 ( 1)...( 1) ( 1) ( 1) ! r r r n n n n r n n n r r r r n r n r n n r r                                   1.21 8! 8 7 6 5 4 560 2! 3! 3! 2 6        1.22 3 2 3 9! 9 8 7 6 5 4 2 3 ( 4) 8 9 64 23,224,320 3! 2! 3! 12               

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    4 Mathematical Statistics, 8E 1.24 Note: If there are 0 turn-ons the first night, 6 turn-ons in four nights can only occur if there are 2 turn-ons on each of the subsequent three nights. Thus, we need to show only that part of the tree following this event.

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    Chapter 1 5 1.25 1.26 (a) (b) 1.27 (a) 5 (b) 4 1.28 1.29 1.30 (a) 6 5 30;   (b) 6 6 36;   1.31 (a) 6; (b) 6 5 30;   (c) 5 4 20   first one fixed; (d) 6 30 20 56   

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    6 Mathematical Statistics, 8E 1.32 (a) 4 5 2 40;    (b) 5 6 3 90    1.33 (a) 5 4 20;   (b) 5 4 3 60    1.34 15 3 14,348,907  1.35 15 14 105 2 1    1.36 (a) 10 9 8 7 5040;     (b) 5040 210 24  1.37 (a) 14 13 91; 2 1    (b) 14 13 12 364 3 2 1      1.38 6! 720  1.39 6! 720 90 2! 2! 2! 8   1.40 5! 120  and 120 2 4! 72    1.41 7! 5040  1.42 (a) 5! 120;  (b) 5! 60 2!  1.43 10! 3628800 50,400 3! 3! 2! 72   and 8! 40320 3360 3! 2! 12   1.44 10! 3628800 1,260 5! 4! 120 24    1.45 8! 40320 280 3! 4! 6 24    1.46 (a) 20 77,520 7        ; (b) 20 184,755 10        (c) 20 20 20 20 1140 190 20 1 1351 17 18 19 20                                 1.47 (a) 7 21 2        ; (b) 4 6 2        ; (c) 3 4 12  

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    Chapter 1 7 1.48 3 7 3 7 3 21 1 7 63 7 70 2 2 3 1                                 1.49 4 7 3 6 35 3 630 2 3 1                       1.50 13 13 13 13 1287 286 286 78 8,211,173,256 5 3 3 2                              1.51 7! 5040 420 3! 2! 12   1.52 10 3 59,049  1.53 5 5 15,625  1.54 12 6 1 17 17 6,188 12 12 5                        1.55 12 1 11 462 6 6                1.56 14 3 1 16 120 14 14                 1.57 2 1 1 1 1 r n n r n n n                     1 10 45 1 2 r n n                 

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    139 Chapter 10 10.1 ( ) i i i i i i E a x a E x a a μ μ            1 1 n i i a     10.2 1 1 2 2 1 2 1 2 ˆ ˆ [ ] , 1 E k k k k k k θ θ θ θ θ       10.3 (2 1)! ( ) ( ) ( ) ( ) ! ! m m x x m h x f x dx f x f x dx m m                          (1/2) (1/2) (2 1)! ( ) 1 ! ! (2 1)! 1 1 1 ! ! 2 2 m m x x m m m h x dx dx m m m x x m m m θ θ θ θ                                                 1 1 ( ) 6( 2 2 h x x x θ θ                     (1/2) (1/2) 1 1 ( ) 6 2 2 E x x x x dx θ θ θ θ                          let u = 1 2 x θ    1 0 1 6 (1 ) 2 u u u du θ θ             10.4 2 / / 6 ( ) 1 8 x x h x e e θ θ         2 / / 0 2 / 3 / 0 0 6 [ ] 1 6 6 5 biased 6 x x x x E x x e e dx x e dx x e dx θ θ θ θ θ θ θ θ                               Use gamma integrals.

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    140 Mathematical Statistics, 8E 10.5   2 2 1 2 2 2 1 1 1 ( ) ( 1 1 n i i i n i E x E x n n n n n μ μ σ σ σ                         10.6 ( ) E x μ  2 var( ) x n σ  2 2 2 2 ( ) E x n σ μ μ    as n   10.7 1 1 1 1 ( 1) ( 1) 2 2 2 2 2 x n E E x n n n n n n θ θ                   when E n θ    , so is asymptotically unbiased 10.8 1 1 1 2 1 1 ( ) ( ) 1 1 ( ) ( 1)( ) ( ) ( ) n y x y y n y n y g y n e e dx n e e n e δ δ δ δ δ                           1 ( ) 1 1 1 1 0 ( ) let 1 ( ) n y nu E y n y e dy u y n u e du n δ δ δ δ δ               The unbiased estimate is 1 1 Y n  1 ( ) as E Y n δ    10.9 1 1 1 1 1 1 1 1 ( ) ( ) n n n y n g y n dx y β β β β β                1 1 1 1 1 1 1 0 1 1 1 1 0 0 ( ) ( ) ( ) (1 ) 1 n n n n n y dy n E Y y y dy u du b u u du n u u du n β β β β β β β β β β β β                 Unbiased estimate is 1 ( 1) n Y  10.10   2 2 1 1 2 2 2 2 1 1 1 1 1 ( ) n n i i i i n n i i x E E x n n n n σ μ σ σ                   

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    Chapter 10 141 10.11 2 2 2 1 1 ( ) ( ) 1 (1 ) ( 1) (1 ) (1 ) biased x x E n E x E x n n n n n n n n n θ θ θ θ θ θ θ θ                               10.12 (a) 1 n  values before n y in 1 1 n y n         ways. 1 1 ( ) n n y n f y k n                for , , n y n k   (b) 1 1 1 ( ) 1 ( 1) see Exercise 1.15 or Theorem 1.11, respectively 1 n n n k k n n n y n y n y y k n n n E Y y k k k n n n n n n k n                                                    1 1 ( 1) 1 1 QED 1 n n n n k E Y k n n n                 10.13 2 2 2 ˆ ˆ ˆ ˆ ( ) var( ) ( ) var( ) E E θ θ θ θ θ     2 2 ( ) since var( ) 0 E θ θ θ     10.14 1 ( ; ) (1 ) x x f x θ θ θ    ( ) E x θ  2 ( ) E x θ  ln ( ; ) ln (1 )ln(1 ) f x x x θ θ θ     ln ( ; ) 1 1 (1 ) f x x x x θ θ θ θ θ θ θ          2 2 2 2 ln ( ; ) 1 1 ( ) (1 ) (1 ) f x E E x θ θ θ θ θ θ θ                      1 (1 ) var x n E n u θ θ           when x is binomial random variable. x n  is minimum variance estimator x n E n n θ θ          unbiased

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    142 Mathematical Statistics, 8E 10.15 ( ; ) ! x e f x x λ λ λ   μ λ  2 σ λ  var( ) x n λ  ( ) unbiased E x λ   ln ln ln ! f x x λ λ    ln 1 f x λ λ     2 2 2 2 2 ln ( ) 2 ( ) 1 2 1 1 f E x E E x λ λ λ λ λ λ λ λ λ                         1 var( ) x nE n λ   x  is minimum variance unbiased estimator 10.16 1 2 ˆ ˆ var( ) 3var( ) θ θ  1 1 2 2 1 2 1 2 ˆ ˆ ( ) 1 E a a a a a a θ θ θ θ θ        2 2 1 2 2 ˆ ˆ var var( ) var( ) i a a θ θ   2 2 2 2 2 2 2 1 2 2 2 2 1 1 2 ˆ ˆ ˆ var 3 var( ) var( ) (3 ) var( ) ˆ [3 (1 ) ]var( ) i a a a a a a θ θ θ θ        2 1 1 1 1 2 6 2(1 )( 1) 1 3 8 2 0 4 4 a a a a a a            10.17 / 1 ( ; ) x f x e θ θ θ   ( ) E x θ  2 2 ( ) 2 E x θ  2 2 σ θ  ( ) unbiased E x θ   2 var( ) x n θ  ln ln x f θ θ    2 2 ln 1 f x x θ θ θ θ θ        2 2 4 2 ln 1 1 ( ) f E E x θ θ θ θ                    2 1 var( ) x x nE n θ    is minimum variance unbiased estimator 10.18 2 2 2 2 ( ) , ( ) , var( ) 1 2 ( 2)( 1) n n n n n n E Y E Y Y n n n n β β β        let 1 n n B Y n    1 ( ) unbiased 1 n n E B n n β β       

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    Chapter 10 143 2 2 2 2 2 ( 1) var( ) ( 2) ( 2)( 1) n n B n n n n n β β        2 2 2 1 1 var( ) 1 ( 2) ln ( ) B n n n f X n nE β β β β              so the Cramèr-Rao inequality is not satisfied. 10.19 (a) ln ( ) 1 ( ) ( ) f x f x f x θ θ      ( ) ln ( ) ( ) f x f x f x θ θ       ln ( ) ( ) 0 f x f x dx θ       (b) 2 2 ln ( ) ln ( ) ln ( ) ( ) ( ) f x f x f x f x f x θ θ θ           2 2 2 ln ( ) ln ( ) ( ) ( ) f x f x f x dx f x dx θ θ                2 2 ln ( ) ln ( ) f x f x E E θ θ                                 10.20 ln ( ) 1 f x x μ μ σ σ           from Example 10.5 2 2 2 ln ( ) 1 f x μ σ     2 2 2 1 1 1 ln ( ) n f x n nE σ σ μ                           10.21 (a) 1 2 [ (1 ) ] (1 ) E wx w x w w μ μ μ       (b) 2 2 2 2 1 2 1 2 var[ (1 ) ] (1 ) wx w x w w n n σ σ      2 2 1 2 2 2(1 )( 1) 0 d w w dw n n σ σ        2 2 2 1 2 2 w σ σ σ   2 2 2 2 1 2 w σ σ σ   10.22 2 2 2 2 1 2 var1 (1 ) w w n n σ σ    1 2 w  2 2 2 2 1 2 1 2 1 var ( ) 4 4 4 n n n σ σ σ σ    

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    144 Mathematical Statistics, 8E     2 2 2 2 2 2 2 1 1 2 2 2 2 2 1 2 1 2 2 2 2 2 2 2 1 2 2 1 1 2 2 2 2 2 2 2 2 1 2 1 2 1 2 var 2 ( ) n n n n σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ                               2 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 2 2 2 1 2 1 2 1 2 2 2 1 2 2 2 2 1 2 ( ) 4 efficiency 1 ( ) ( ) 4 4 ( ) n n n n σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ           10.23 2 2 2 2 1 2 var (1 ) w w n n σ σ    2 2 1 2 2 2(1 ) 0 d w w dw n n σ σ     1 2 1 w w n n   1 1 2 n w n n   10.24 For 1 2 w  2 2 2 1 2 1 2 1 1 1 1 var 4 4 4 n n n n σ σ σ            For 1 1 2 n w n n   2 2 2 2 1 2 1 2 1 1 2 2 2 2 1 2 2 1 2 1 2 var ( ) ( ) n n n n n n n n n n n n n n σ σ σ σ                      Efficiency = 2 1 2 1 2 2 2 1 2 1 2 4 ( ) 1 1 4 n n n n n n n n σ σ           10.25 2 2 2 2 1 2 3 2 1 1 1 3 var 4 16 4 16 8 x x x σ σ σ σ             2 var( ) 3 x σ  Efficiency = 2 2 8 3 3 9 8 σ σ 

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