Probability, Statistics, and Stochastic Processes, 2nd Edition

Appendix B

Answers to Selected Problems

Chapter 1

1. (a) {3, 4, . . ., 18} (b) [0, 1] × [0, 1] (c) {M, F} × {0, 1, 2, . . . }

(d) {(i, j): 1 ≤ i < j ≤ 10} (e) [0, 1].

2. S = {rr, rs, sr, ss}

3. (c) and (e)

4. (a) (A ∩ B^c ∩ C^c) ∪ (A^c ∩ B ∩ C^c) ∪ (A^c ∩ B^c ∩ C) (b) A^c ∩ B^c ∩ C^c

(c) A ∪ B ∪ C

5. (a) B₁ (b) (c) B₁ ∩ B₂ ∩ B₃ ∩ B₄

(d)

(e)

6. 0.6

7. (a) P(B) = 0.4 (b, c) P(A ∩ B^c) = 0.1 (d) P(A^c) = 0.7 (e) P(B^c) = 0.6

(f) P(A^c ∩ B^c) = 0.5 (which also equals P((A ∪ B)^c)).537

8. (a) 0.5 − p (b) 1 − 2p (c) p

10. (b)

12. (a) 3/8 (b) 1/8 (c) 7/8 (d) 1/8 (e) 3/8

16. (a) 0.5 + 0.33 + 0.2 − (0.16 + 0.2 + 0.08) + 0.08 = 0.67

(b) P(A_i)+ P(A_j) + P(A_k) − (P(A_lcm(i,j)) + P(A_lcm(i,k)) + P(A_lcm(j,k))) + P(A_lcm(i,j,k)) (see Example 1.10)

17. (a) (7.1 + 15 − 0.75)/143 = 0.149

(b) (7.1 + 15 + 10 − (0.75 + 0.5 + 0) + 0)/143 = 0.216

(c) (71 + 15 + 71 + 1 − (7.5 + 35.5 + 0 + 7.5 + 0 + 0) + (3.75 + 0 + 0 + 0) − 0)/143 = 0.778

19. (a) (b)

20. (a) 23 × 22 × 21/23³ = 0.87 (b) 10 × 9 × 8/10³