Problems 1.3

1. (a)Yes; (b) yes; (c) no.  2. (a) Yes; (b) no; (c) no.

6. (a) 0.9; (b) 0.05; (c) 0.95.  7.1/16.   8. .

PROBLEMS 1.4

3.   4. 352146  5. !

6. 1–₇P_5/75  8.   9.

12. (a)  (b)  (c)

  (d)  (e)

  (f)  (g)

  (h)  (i)

Problems 1.5

3.   4. p/(2–p)

5.   6.

10.  11. (a) 1/4; (b) 1/3  12. 0.08

13. (a) 173/480 (b) 108/173; 15/173  14. 0.0872

PROBLEMS 1.6

1.  4.

12. For any two disjoint intervals , where ℓ(I) = length of interval I.

13. (a)

(b) 22/45

(c)

PROBLEMS 2.2

3. Yes; yes

4. ϕ;{(1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,2,1,1,1), (2,1,1,1,1)};{(6,6,6,6,6)};

{(6,6,6,6,6), (6,6,6,6,5), (6,6,6,5,6), (6,6,5,6,6), (6,5,6,6,6), (5,6,6,6,6)}

5. Yes;

PROBLEMS 2.3

1.

3. (a) Yes; (b) yes; (c) yes; yes

PROBLEMS 2.4

1. (1−p)ⁿ⁺¹−(1−p)^N+1, N ≥ n

2. (b) ; (c) 1/x²; (d)

3. Yes;

4. Yes; for

6. for for

8. (c), (d), and (f)

9. Yes; (a) for ; (b) ;

 (c) for , and ;

 (e)

10. If , then

Problems 2.5

2.

4. ;

10. for for

12. (a) for , and 1 for ;

 (b) ;

 (c) .

   if .

PROBLEMS 3.2

3.

9. , where

10. Binomial:

  Poisson: .

PROBLEMS 3.3

1. (b) .

6. f(θs)/f(θ); f(θe^t)/f(θ).

PROBLEMS 3.4

3. For any take .

5.

PROBLEMS 4.2

1. No  4. 1/6; 0.   7. Marginals negative binomial, so also conditionals.

8.

9. .

10. .

14.   15. 1/24; 15/16.   17. 1/6.

PROBLEMS 4.3

3. No; Yes; No.  10. .

11. .

PROBLEMS 4.4

2. (b) .

6.

7.

11. .

13.

PROBLEMS 4.5

2. . 3. dependent.

15. for ; no.

18.

21. If U has PDF f, then for ;.

PROBLEMS 4.6

1. where Φ is the standard normal DF.

2. (a) . 3.  4. .

6. 4/9. 7. (a) 1; (b) 1/4. 8. .

PROBLEMS 4.7

5. (a) ; (b) .

PROBLEMS 5.2

5.

9.

PROBLEMS 5.3

2. (a) .

13.

22. .

27. (a) t/α²; (c) ; (d) .

29. (b) .

Problems 5.4

1. (a) ; (b) ; (c) 0.3191.

4. . 6. . 7. .

PROBLEMS 6.2

1.

2.

PROBLEMS 6.3

1. .

.

.

PROBLEMS 6.4

.

.

.

PROBLEMS 6.5

1. 2.  3.

PROBLEMS 6.6

3. .

4.

PROBLEMS 7.2

1. No. 2. Yes

3. .

4. .

9.  12. No

13. (a) .

  (b) .

  (c) .

20. (a) Yes; No (b) Yes; No.

PROBLEMS 7.3

3. Yes;

5. (a) as ; no. (b) M_n(t) diverges as

(c) Yes (d) Yes (e) ; no.

Problems 7.4

1. (a) No; (b) No; 2. No; 3. For . 7. (a) Yes; (b) No.

Problem 7.5

4. Degenerate at β. 5. Degenerate at 0.

6. .

Problem 7.6

1. (b) No; (c) Yes; (d) No.

2. . 3. . 4. 163. 8. 0.0926; 1.92

Problem 7.7

1. .

2. .

Problem 8.3

2. No. 7. . 9. No. 10. No.

11. (b) X_(n); (e) ; (g)  (h) X(₍₁₎, X₍₂₎, …, X_(n)).

PROBLEMS 8.4

2. .

3. ; 4. No; 5. No.

6.

9.  11. (a) NX/n; (b)No.

12. if , and 1 if .

13. (a) With  (b)  (c) (1–1/n)^t;

(d) .

14. With .

15. With

Problems 8.5

1. (a), (c), (d) Yes; (b) No. 2. 0.64761/n².

3. . 5.

PROBLEMS 8.6

2.  3. .

4.

5.

PROBLEMS 8.7

1. (a) med(X_j); (b)X₍₁₎; (c)  (d) .

2.

3. .

4. (a)  (b) ,

  .

5. . 6.

8. (a) ; (b) X_(M).

9.

11. , 13. . 15. .

16. .

PROBLEMS 8.8

2. (a) ; (b) . 3.  5. X/n.

6. . 8. .

PROBLEMS 8.9

5. (c)

10.

PROBLEMS 9.2

1. 0.019, 0.857. 2. .

5. .

Problems 9.3

1. oterwise.

4. . 5. .

11. If , and if , then if .

12. .

PROBLEMS 9.4

1. (a), (b), (c), (d) have MLR in

4. Yes. 5. Yes; yes.

PROBLEMS 9.5

1. otherwise,

2. . Choose k from .

PROBLEMS 9.6

3. if .

PROBLEMS 10.2

2. in sample, or . 3. .

4. . 5. (a) ; (b) .

6. . 7. (a) ; (b) .

11. . 12. .

PROBLEMS 10.3

1. Reject at . 3. Do not reject at 0.05 level.

4. Reject H₀ at . 5. Reject at 0.10 but not at 0.05 level.

7. Do not reject H₀ at . 8. Do not reject H₀ at .

10. . 12. .

PROBLEMS 10.4

1. , reject H₀ at . 2. , do not reject H₀.

5. . 6. Reject H₀ at . 7. Reject H₀. 8. Reject H₀.

PROBLEMS 10.5

1. Do not reject .

3. Do not reject H₀ at . 4. Do not reject H₀.

PROBLEMS 10.6

2. (a) if if otherwise;

(b) Minimax rule rejects H₀ if or 5, and with probability 1/16 if ;

(c) Bayes rule rejects H₀ if .

Problems 11.3

1. (77.7, 84.7). 2. . .

. 10. [α^1/nN]

.

12. Choose k from .

15. Posterior .

, where Φ is standard normal DF.

PROBLEMS 11.4

.

, choose a, b from , where is the PDF of χ²(v) RV.

3. , choose a, b from and .

PROBLEMS 11.5

1. .

2. (2∑X_i/λ₂, 2∑X_i/λ₁), where λ1,λ2 are solutions of λ1f2nα(λ1) = λ2f2nα(λ2) and P(1) = 1−α, fv is χ2(v) PDF.

3. .

5. (α^1/nX₍₁₎, X₍₁₎).

8. Yes

PROBLEMS 12.3

4. Reject .

8.

10. (a) , (b) ; (b) , reject H₀.

PROBLEMS 12.4

2. . 3. Reject at but not at .

4. , reject at but not at 0.01.

5. . 6. .

PROBLEMS 12.5

4. ; reject H₀ at , not at 0.01.

5. .

Problems 12.6

2. Reject H₀ if .

4. ;

.

5. Cities 3 227.27 4.22

Auto 3 3695.94 68.66

Interactions 9 9.28 0.06

Error 16 287.08

PROBLEMS 13.2

1. d is estimable of degree 1; (number of xi’s in A)/n.

2. (a) ; (b) .

3. (a) ∑X_iY_i/n; (b) .

PROBLEMS 13.3

3. Do not reject H₀. 7. Reject H₀. 10. Do not reject H₀ at 0.05 level.

11. , do not reject H₀.

12. (Second part) , do not reject H₀ at .

PROBLEMS 13.4

1. Do not reject H₀. 2. (a) Reject; (b) Reject.

3. , reject H₀. 5. , do not reject H₀.

7. , reject; or 12, do not reject at .

PROBLEMS 13.5

1. Reject H₀ at . 4. Do not reject H₀ at .

9. (a) ; (b) ; (c) Reject H₀ in each case.

PROBLEMS 13.6

1. (a) 5; (b) 8. 3. .

4. .

PROBLEMS 13.7

1. (c) ; ratio = 1 if for .

2. Chi-square test based on (c) is not robust for departures from normality.