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. .
3. 4. 352146 5. !
6. 1–7P5/75 8. 9.
12. (a) (b) (c)
(d) (e)
(f) (g)
(h) (i)
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
1. 4.
12. For any two disjoint intervals , where ℓ(I) = length of interval I.
13. (a)
(b) 22/45
(c)
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;
1.
3. (a) Yes; (b) yes; (c) yes; yes
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
2.
4. ;
10. for for
12. (a) for , and 1 for ;
(b) ;
(c) .
if .
3.
9. , where
10. Binomial:
Poisson: .
1. (b) .
6. f(θs)/f(θ); f(θet)/f(θ).
3. For any take .
5.
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.
3. No; Yes; No. 10. .
11. .
2. (b) .
6.
7.
11. .
13.
2. . 3. dependent.
15. for ; no.
18.
21. If U has PDF f, then for ;.
1. where Φ is the standard normal DF.
2. (a) . 3. 4. .
6. 4/9. 7. (a) 1; (b) 1/4. 8. .
5. (a) ; (b) .
5.
9.
2. (a) .
13.
22. .
27. (a) t/α2; (c) ; (d) .
29. (b) .
1. (a) ; (b) ; (c) 0.3191.
4. . 6. . 7. .
1.
2.
1. .
.
.
.
.
.
1. 2. 3.
3. .
4.
1. No. 2. Yes
3. .
4. .
9. 12. No
13. (a) .
(b) .
(c) .
20. (a) Yes; No (b) Yes; No.
3. Yes;
5. (a) as ; no. (b) Mn(t) diverges as
(c) Yes (d) Yes (e) ; no.
1. (a) No; (b) No; 2. No; 3. For . 7. (a) Yes; (b) No.
4. Degenerate at β. 5. Degenerate at 0.
6. .
1. (b) No; (c) Yes; (d) No.
2. . 3. . 4. 163. 8. 0.0926; 1.92
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)).
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
1. (a), (c), (d) Yes; (b) No. 2. 0.64761/n2.
3. . 5.
2. 3. .
4.
5.
1. (a) med(Xj); (b)X(1); (c) (d) .
2.
3. .
4. (a) (b) ,
.
5. . 6.
8. (a) ; (b) X(M).
9.
11. , 13. . 15. .
16. .
2. (a) ; (b) . 3. 5. X/n.
6. . 8. .
5. (c)
10.
1. 0.019, 0.857. 2. .
5. .
1. oterwise.
4. . 5. .
11. If , and if , then if .
12. .
1. (a), (b), (c), (d) have MLR in
4. Yes. 5. Yes; yes.
1. otherwise,
2. . Choose k from .
3. if .
2. in sample, or . 3. .
4. . 5. (a) ; (b) .
6. . 7. (a) ; (b) .
11. . 12. .
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. .
1. , reject H0 at . 2. , do not reject H0.
5. . 6. Reject H0 at . 7. Reject H0. 8. Reject H0.
1. Do not reject .
3. Do not reject H0 at . 4. Do not reject H0.
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 .
1. (77.7, 84.7). 2. . .
. 10. [α1/nN]
.
12. Choose k from .
15. Posterior .
, where Φ is standard normal DF.
.
, choose a, b from , where is the PDF of χ2(v) RV.
3. , choose a, b from and .
1. .
2. (2∑Xi/λ2, 2∑Xi/λ1), where λ1,λ2 are solutions of λ1f2nα(λ1) = λ2f2nα(λ2) and P(1) = 1−α, fv is χ2(v) PDF.
3. .
5. (α1/nX(1), X(1)).
8. Yes
4. Reject .
8.
10. (a) , (b) ; (b) , reject H0.
2. . 3. Reject at but not at .
4. , reject at but not at 0.01.
5. . 6. .
4. ; reject H0 at , not at 0.01.
5. .
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
1. d is estimable of degree 1; (number of xi’s in A)/n.
2. (a) ; (b) .
3. (a) ∑XiYi/n; (b) .
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 .
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 .
1. Reject H0 at . 4. Do not reject H0 at .
9. (a) ; (b) ; (c) Reject H0 in each case.
1. (a) 5; (b) 8. 3. .
4. .
1. (c) ; ratio = 1 if for .
2. Chi-square test based on (c) is not robust for departures from normality.