Sections 6.6, 6.7 Class Exercises
Math 400 - Actuarial Models
NAME:_____________________________
1. The life table on the right is defined by the
values of lx .
(a) With the UDD (linear) assumption, find each
of the following:
l 6.75
0.75 p 6
0.75 q 6
 6.75
0.75 p 6
 6.75
1
x
lx
_____________
0 1000000
1
968377
2
910557
3
835683
4
747018
5
646447
6
535242
7
414338
8
284458
9
146185
10
0
Sections 6.6, 6.7 Class Exercises
1.-continued
(b) With the exponential (constant force) assumption, find each of the
following:
l 6.75
0.75 p 6
0.75 q 6
 6.75
0.75 p 6
 6.75
2
Sections 6.6, 6.7 Class Exercises
1.-continued
(c) With the hyperbolic assumption, find each of the following:
l 6.75
0.75 p 6
0.75 q 6
 6.75
0.75 p 6
 6.75
3
Sections 6.6, 6.7 Class Exercises
1.-continued
(d) With the UDD (linear) assumption, find the median of the lifetime
distribution.
(e) With the exponential (constant force) assumption, find the median of
the lifetime distribution.
2. Prove that with the UDD (linear) assumption, if t > 0, u > 0, and
t + u < 1, then u | t q x = t q x .
4
Sections 6.6, 6.7 Class Exercises
3. Let T0 be a lifetime random variable, and random variables Tx , Kx , and
Rx are as previously defined.
(a) Find the distribution of Rx with the UDD (linear) assumption.
(b) Let k be a nonnegative integer and 0  s < 1. Find a formula for each
of P[Kx = k], P[Rx  r], and P[{Kx = k}  {Rx  r}] with the UDD (linear)
assumption. From these formulas, draw a conclusion about the relationship
between random variables Kx and Rx under the UDD (linear) assumption.
5
Sections 6.6, 6.7 Class Exercises
4. Displayed below is an excerpt from a select and ultimate table with a
five-year select period. The radix is 1,000,000.
x
l[x]
l[x] + 1
l[x] + 2
l[x] + 3
l[x] + 4
lx + 5
x+5
40
676154
674698
670339
663086
652949
639939
45
41
668876
667424
663075
655840
645726
632746
46
42
661617
660168
655829
648610
638520
625568
47
43
654374
652929
648600
641397
631329
618406
48
44
647148
645706
641387
634200
624154
611258
49
45
639939
638500
634191
627019
616994
604125
50
46
632746
631310
627010
619853
609849
597007
51
47
625568
624136
619844
612702
602719
589902
52
48
618406
616976
612694
605566
595603
582811
53
49
611258
609832
605558
598444
588500
575733
54
50
604125
602702
598436
591337
581412
568668
55
51
597007
595586
591328
584243
574336
561617
56
52
589902
588484
584234
577162
567274
554578
57
53
582811
581395
577154
570094
560224
547551
58
54
575733
574320
570087
563040
553187
540536
59
55
568668
567258
563032
555998
546162
533533
60
56
561617
560209
555990
548968
539150
526542
61
57
554578
553172
548961
541951
532149
519562
62
58
547551
546148
541944
534945
525160
512594
63
59
540536
539135
534938
527952
518182
505636
64
60
533533
532135
527945
520969
511215
498690
65
(a) Find 3 q 48 .
6
Sections 6.6, 6.7 Class Exercises
4.-continued
(b) Find 3 q [48] .
(c) Find 5 q 48 .
(d) Find 5 q [48] .
(e) Using the UDD assumption, find 0.8 q [48] + 0.4 .
7