Answers to Supplementary Exercises

Chapter 1 - Survival Models

1. \(\quad\) (a) This function does satisfy the conditions for a survival function: it can be demonstrated that it is non-increasing (by proving that \(f(b) \leq f(a)\), \(\forall b>a\)); \(s(0)=1\) and \(s(100)=0\); and continuous (by proving that \(\lim_{x\to c} s(x)=s(c)\); (b) \(\omega = 100\)); (c) \(0.76375\); (d) \(0.23196\)

2. \(\quad\) (a) \(0.75\); (b) \(0.27\); (c) \(0.68235\); (d) \(0.5625\)

3. \(\quad\) (d) \(\omega = 125\)

4. \(\quad\) (a) This function does satisfy the conditions for a survival function: it can be demonstrated that it is non-increasing (by proving that \(f(b) \leq f(a)\), \(\forall b>a\)); \(s(0)=1\) and \(\lim_{x\to \infty} s(x)=0\); and continuous (by proving that \(\lim_{x\to c} s(x)=s(c)\))

5. \(\quad\) (a) \(0.5\); (b) \(0.3466\); (c) \(0\); (d) \(0.5\)

6. \(\quad 14\)

7. \(\quad 16.36\)

8. \(\quad 0.18182\)

9. \(\quad \dfrac{4}{9}\)

10. \(s(x)\)	\(F(x)\)	\(f(x)\)	\(\mu_x\)
    \(1-\dfrac{x}{100}\)	\(\dfrac{x}{100}\)	\(\dfrac{1}{100}\)	\((100-x)^{-1}{,}\) \(0 \leq x \leq 100\)
    \(e^{-x}{,}\) \(x \geq 0\)	\(1-e^{-x}\)	\(e^{-x}\)	\(1\)
    \(\dfrac{1}{1+x}\)	\(1-(1+x)^{-1}{,}\) \(x \geq 0\)	\(\dfrac{1}{(1+x)^2}\)	\(\dfrac{1}{1+x}\)
    \(\dfrac{(100-x)^2}{10{,}000}\)	\(\dfrac{x(200-x)}{10{,}000}\)	\(\dfrac{100-x}{5\,000}{,}\) \(0 \leq x \leq 100\)	\(\dfrac{2}{100-x}\)

    \(\\\)

11. \(\quad 2.773\)

12. \(\quad 0.00675\)

13. \(\quad z(n-x)-z(m-x)\)

14. \(\quad 27.55\)

15. \(\quad\) (a) \(50\); (b) \(208.3\overline{3}\); (c) \(50\)

16. \(\quad 0.41803\)

17. \(x\)	\(l_x\)	\(p_x\)	\(q_x\)	\(d_x\)
    \(0\)	\(100\,000\)	\(0.99450\)	\(0.00550\)	\(550\)
    \(1\)	\(99\,450\)	\(0.99346\)	\(0.00654\)	\(650\)
    \(2\)	\(98\,800\)	\(0.99241\)	\(0.00760\)	\(750\)

\(\\\)

18. \(x\)	\(q_x\)	\(l_x\)	\(d_x\)
    \(0\)	\(0.011\)	\(1\,000\,000\)	\(11\,000\)
    \(1\)	\(0.005\)	\(980\,000\)	\(4\,945\)
    \(2\)	\(0.003\)	\(984\,055\)	\(2\,952\)

\(\\\)

19. \(\quad\) (a) \(0.70355\); (b) \(0.29656\); (c) \(0.02282\); (d) \(0.17768\)

20. \(\quad 1\)

21. \(\quad 0.75\)

22. \(\quad 10\log{2}\)

23. \(\quad 1.5\)

24. \(\quad 26{,}750\)

25. \(\quad 0.992\)

26. \(\quad 0.4\)

27. \(\quad\) (a) \(0.01176\); (b) \(42.5\)

28. \(\quad 15\,000\)

29. \(\quad 0.17324\)

30. \(\quad 126.28\)

Chapter 2 - Life Insurance Models

1. \(\quad 5\)

2. \(\quad 0.62073\)

3. \(\quad 0\)

4. \(\quad 0.04879\)

5. \(\quad 10\)

6. \(\quad 0.43047\)

7. \(\quad 85\)

8. \(\quad 0.25170\)

9. \(\quad 17{,}442\)

10. \(\quad 416.6\bar{6}\)

11. \(\quad 26{,}600\)

12. \(\quad\) (a) \(1{,}419\); (b) \(10{,}514\)

13. \(\quad 49{,}122\)

14. \(\quad 0.05\)

15. \(\quad 0.76204\)

16. \(\quad 6\)

17. \(\quad 3{,}241.38\)

18. \(\quad 5{,}597\)

19. \(\quad 58{,}739\)

20. \(\quad 111{,}612.50\)

21. \(\quad 353.45\)

22. \(\quad 0.96129\)

23. \(\quad 6{,}952.38\)

24. \(\quad 20{,}876.49\)

25. \(\quad\) –

26. \(\quad 0.27733\)

27. \(\quad 0.00542\)