Homework Answers
- Homework I: (1) Q/P=5/16. (2) expected value=1. (3) relative
security loading =0.59. (4) E[D]=0.65.
(5) premium=0.123. (6) alpha=2.
(7) probability=0.995. (8) security loading for nonsmoker=0.0697.
- Homework II: (1) probability=0.606. (2) k=4. (3)
variance=0.5. (4) probability=0.25.
(5) variance=0.44. (6) mean+standard
deviation=56.27. (7) expected compensation= 55,317. (8) All of (a), (b), (c)
correct.
- Homework III: (1) probability=0.238. (2) expected profit = loss of
0.67 percent. (3) standard deviation =2792. (4) F(4)=2.5exp(-1).
(5) minimum
size= 55,602. (6) expected aggregate losses= 7.725. (7) probability= 0.1587.
(8) expected aggregate losses= 1.43.
- Homework IV: (1) number of claims=4029. (2) number of claims =1000.
(3) number of claims = 1883. (4) loss cost= 113.5. (5) credibility= 0.857.
(6) credibility= 0.866. (7) p(x)=exp(-x)/[1-exp(-1)], x in [0,1].
(8) (a) 4/9; (b) P( die=A|1 appears on roll)=1/2,
P( die=A|0 appears on roll)=1/5,
P( die=B|1 appears on roll)=9/24,
P( die=B|0 appears on roll)=3/10,
P( die=C|1 appears on roll)=1/8,
P( die=C|0 appears on roll)=1/2;
(c) Expected value of 2nd roll, given 1 appears on 1st roll= 13/24,
Expected value of 2nd roll, given 0 appears on 1st roll= 11/30;
(d) 4/9; (e) same by Bayes theorem.
- Homework V: (1) (a) 0.03, (b) 11.97, (c) 399.
(2) credibility
annual frequency for person with 0 claims=5/24, with 1 claim=1/4, 2
claims=7/24, 3 claims=1/3, 4 claims=3/8, 5 claims=5/12.
(3) 5.14. (4) 5.72.
(5) (y+alpha)/(beta+N). (6) 7/9. (7) 44.69. (8) 5/84.
- Homework VI: (1) 498. (2)0.0236. (3) 25.87. (4) 0.234.
(5) 800.
(6) 41 minutes. (7) 0.21. (8) (a) 1/9, (b) 5/9.
- Homework VII: (1) (a) 3/4, (b) 7.5 minutes.
(2) probability=[1+2exp(-4)]/3. (3) 14.17 minutes. (4) 11,946.
(5) adjustment coefficient for I is R, for II it is R/2, and for III it is R/2.
(6) 139/920. (7) security loading=8.99. (8) probability=0.6.
- Homework VIII: (1) surplus=ln(5).
(2) mgf(t)=t*theta/[1+(1+theta)t-exp(t)]. (3) probability=11/32. (4) expected
benefits=41,960.
(5) adjustment coefficient=6/11.
(6) reinsurance premium=1/3.
(7)probability of ruin=1/16. (8) alpha=1/3.
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