ANSWERES TO SELECTED REVIEW EXERCISES

13. S has countable number of elements.

14. S has uncountable number of elements.

16. (n 1)(n 2) 1 2 n+1 .

20. 3n 1 n+1 .

Answers to Selected Problems

CHAPTER 2

3 15. (a) 4 2 4 + 3 4

5 16 and (b)

Probability and Mathematical Statistics

2. 2k+1 k+1 .

3. p 3 1 2 .

4. Mode of X = 0 and median of X = 0.

8. f(2) = 0.5, f(3) = 0.2, f(⇡) = 0.3.

9. f(x) = 1 3 6 x x e .

11. a = 500, mode = 0.2, and P (X

17. R X = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12};

f (2) = 1 , f (3) = 2 , f (4) = 3 , f (5) = 4 , f (6) = 5 36 6 36 36 36 36 , f (7) = 36 , f (8) =

5 , f (9) = 4 1

36 , f (10) = 3 36 , f (11) = 36 2 36 , f (12) = 36 .

18. R X = {0, 1, 2, 3, 4, 5};

f (0) = 59049 10 5 , f (1) = 32805 10 5 , f (2) = 7290 10 5 , f (3) = 810 10 5 , f (4) = 45 10 5 , f (5) = 1 10 5 .

19. R X = {1, 2, 3, 4, 5, 6, 7};

f (1) = 0.4, f (2) = 0.2666, f (3) = 0.1666, f (4) = 0.0952, f (5) = 0.0476, f(6) = 0.0190, f(7) = 0.0048.

20. c = 1 and P (X = even) = 1 4 .

21. c = 1 2 3 , P (1  X  2) = 4 .

22. c = 3 2 1 and P X  2 = 3 16 .

Answers to Selected Problems

3. (c) 0.25, (d) 0.75, (e) 0.75, (f) 0.

4. (a) 3.75, (b) 2.6875, (c) 10.5, (d) 10.75, (e) 71.5.

5. (a) 0.5, (b) ⇡, (c) 3 10 ⇡.

12. a = 4h p 3 , E(X) = p 2 , V ar(X) = 1 2 ⇥ 3 4 ⇡ ⇤ h ⇡ h 2 ⇡ .

13. E(X) = 7 4 , E(Y ) = 7 8 .

16. M(t) = 1 + 2t + 6t 2 + · · ·.

21. E(X) = 3, V ar(X) = 2.

23. c = E(X).

24. F (c) = 0.5.

25. E(X) = 0, V ar(X) = 2.

28. a = 5 and b = 34 or a = 5 and b = 36.

31. 1p 1 p ln p.

Probability and Mathematical Statistics

13. n 2 2 n+1 3n+2 .

3k )( k )

( 10 3 ) , 0  k  3.

17. 1 1 e 2 .

18. x1 1 3 5 2 x3 6 6 .

26. e 4 4 1 .

Answers to Selected Problems

CHAPTER 6

1. f(x) = e x

0 < x < 1.

2. Y ⇠ UNIF (0, 1).

1 1 ( ln w µ 3. f(w) = 2 p

18. g(y) =

24. ln(X) ⇠ V

29. Y ⇠ GBET A(↵, , a, b).

32. (i) 1 p ⇡, (ii) 2 1 , (iii) 1 p

2 4 ⇡, (iv) 2 .

33. (i) 1 , (ii) (100) 13 5! 7! , (iii) 180 1

36. E(X n )=✓ n (n+↵) (↵) .

Probability and Mathematical Statistics

CHAPTER 7

1. f 1 (x) = 2x+3 21 , and f 2 (y) = 3y+6 21 .

36 if 1 < x < y = 2x < 12

2. f(x, y) = 2 36 if 1 < x < y < 2x < 12

2 0 (e otherwise. 1)(e 1)

12. f(yx) = 1+

14. f(yx) = 2x if 0 < y < 2x < 1

0 otherwise.

16. g(w) = 2e w 2e 2w .

⇣

⌘ 6w 2

17. g(w) = 1 w 3

✓ 3 ✓ 3 .

21. No.

22. Yes.

26. x e x .

Answers to Selected Problems

2. Cov(X, Y ) = 0. Since 0 = f(0, 0) 6= f 1 (0)f 2 (0) = 1 4 , X and Y are not

independent.

3. p 1 8 .

4. (1 4t)(1 6t) 1 .

5. X + Y ⇠ BIN(n + m, p).

6. 1 2 2 X +Y 2 ⇠ EXP (1).

7. M(s, t) = t e 1

10. Cov(X, Y ) = 0. No.

11. a = 6 8 and b = 9 8 .

12. Cov = 45 112 .

13. Corr(X, Y ) = 1 5 .

n [1 + (n 1)⇢].

Probability and Mathematical Statistics

7. 1 2+3y 28y 3 3 1+2y 8y 2 .

8. 3 2 x.

9. 1 2 y.

10. 4 3 x.

12. 15 1 ⇡ .

13. 12 1 (1 x) 2 .

14. 12 1 1x 22 .

8 9 6y

< 7 for 0  y  1

15. f 2 (y) =

3(2 y) 2 and V ar X|Y = 2 = 72 : .

7 if 1  y  2

19. x 6 + 5 12 .

20. x 2 + 1.

Answers to Selected Problems

1. g(y) =

2. g(y) =

3. g(y) =

4. g(z) =

5. g(z, x) =

6. g(y) =

0 otherwise.

8 z 3 z 2

250 + > z 15000 25 for 0  z  10

7. g(z) = 2 8 2z z

> 15 25 250 15000 for 10  z  20

8. g(u) =

0 otherwise.

9. h(y) = 3z 2 216 2z+1 , z = 1, 2, 3, 4, 5, 6.

8 q

< 4h 3 2z m 2h2z p ⇡ m e m for 0  z < 1

10. g(z) =

11. g(u, v) =

0 otherwise.

( 2u

(1+u) 3 if 0  u < 1

12. g 1 (u) =

0 otherwise.

Probability and Mathematical Statistics

8 5 [ 9v 3 5u 2 v+3uv 2 +u 3

13. g(u, v) =

14. g(u, v) =

: 0 otherwise. ( 2u 3 if 1  u < 1

19. f(w) =

20. BIN(2n, p)

21. GAM(✓, 2)

22. CAU(0)

23. N(2µ, 2 2 )

4 (2 |↵|) if |↵|  2

2 ln(| |) if | |  1

24. f 1 (↵) =

f 2 ()=

: 0 otherwise,

0 otherwise.

Answers to Selected Problems

CHAPTER 11

Probability and Mathematical Statistics

CHAPTER 12

Answers to Selected Problems

16. X has a degenerate distribution with MGF M(t) = e 1 2 t .

17. P OI(1995 ).

3 20. f(x) = 3x 60

✓ 1e ✓ e ✓ for 0 < x < 1.

21. X (n+1) ⇠ Beta(n + 1, n + 1).

Probability and Mathematical Statistics

2. (3); the MGF of X 2 1 X 2 is M(t) = p 1 1 4t 2 .

3. t(3).

4. f(x (x1+x2+x3)

7. M(t) = p

1 (1 2t)(1 4t)(1 6t)(1 8t) .

n 2 2(n 1).

12. 2 (2n).

13. t(n + p).

14. 2 (n).

17. 2 n 2 .

19. 2 (2n 2).

Answers to Selected Problems

11. ˆ ↵ = 3.534 and ˆ = 3.409.

13. 1 3 max{x 1 ,x 2 , ..., x n }.

14. 1 max {x 1 ,x 2 ,...,x n } .

19. 1+ 5 ln(2) .

20. ¯ X

1+ ¯ X .

21. X ¯ 4 .

23. n

|X i µ |

Probability and Mathematical Statistics

32. b 2(x ✓ is obtained by solving numerically the equation i ✓)

P n

i=1 1+(x i ✓) 2 = 0.

33. b ✓ is the median of the sample.

34. n .

35. (1 p) p n 2 .

36. b ✓ = 3 X.

37. b ✓= 50 30 X.

Answers to Selected Problems

CHAPTER 16

2 cov(T ,T 1. b = )

1 2 + 2 1 2cov(T 2 1 ,T 2 .

✓= 2. b |X| , E( |X| ) = ✓, unbiased.

15. X (1) , and sufficient.

16. X (1) is biased and X

1 is unbiased. X (1) is efficient then X 1.

X n

17. ln X i .

i=1

X n

18. X i .

i=1

X n

19. ln X i .

i=1

22. Yes.

23. Yes.

Probability and Mathematical Statistics

24. Yes.

25. Yes.

26. Yes.

Answers to Selected Problems

CHAPTER 17

7. The pdf of Q is g(q) = nq ne

h ⇣ The confidence interval is X (1) 1 n ln 2 1 2

8. The pdf of Q is g(q) = 2 e 2 q

The confidence interval is X 1

9. The pdf of Q is g(q) = nq n1

if 0 < q < 1

0 otherwise.

h ⇣ ⌘i The confidence interval is X 1 n ln 2 (1) 1 ↵ ,X (1) n ln 2 2↵ .

⇢

10. The pdf g(q) of Q is given by g(q) = nq n1

The confidence interval is 2 1 n

11. The pdf of Q is given by g(q) = n (n

13. b ✓ z ↵ 2 b ✓+1 p ,b ✓+z ↵ 2 b n ✓+1 p n , where b ✓= 1+ P n n

16. X (n) z 2 (n+1) n+2 ,X (n) +z 2 (n+1) p n+2 .

17. 1 X 1 X

4 X z 2 ↵ 8 p n , 4 X+z ↵ 2 8 p n .

Probability and Mathematical Statistics

CHAPTER 18

1. ↵ = 0.03125 and = 0.763.

2. Do not reject H o .

X 7 (8 ) x e 8

3. ↵ = 0.0511 and ( ) = 1

4. ↵ = 0.08 and = 0.46.

5. ↵ = 0.19.

6. ↵ = 0.0109.

7. ↵ = 0.0668 and = 0.0062.

9. C = {(x 1 , ..., x 10 ) | x 0.3}.

10. C = {x 2 [0, 1] | x 0.829}.

11. C = {(x 1 ,x 2 )|x 1 +x 2 5}.

12. C = {(x 1 , ..., x 8 ) | x x ln x  a}.

13. C = {(x 1 , ..., x n ) | 35 ln x x  a}.

14. C = (x , ..., x

x 1 5 5 )| 2x 2 a.

15. C = {(x 1 ,x 2 ,x 3 ) | |x 3| 1.96}.

1 o

16. C = (x 1 ,x 2 ,x 3 )|xe 3 x a .

17. C = (x 1 ,x 2 , ..., x n )|

20. C = {(x 1 ,x 2 ,x 3 ) | x 12.04}.

21. ↵ = 1 16 and = 255 256 .

22. ↵ = 0.05.

Answers to Selected Problems

i=1 (n i 10) 2 63.43.

10. 25.

ANSWERES TO SELECTED REVIEW EXERCISES