| Previous | An Introduction to the Theory of Life Contingencies (1931) | Next |
|
3. |
|
SCHEDULE OF THE CONNECTIONS BETWEEN Ax, ax, Px AND d |
| ||||||
|
|
|
|
future life time of man aged x |
year of death |
| ||||
|
|
|
|
|
.. .. .. .. .. .. .. .. .. .. .. .. |
|
|
| ||
|
|
x |
x+1 |
x+2 |
|
|
|
|
| |
|
ax = the series |
1 |
1 |
1 |
|
1 |
1 |
|
.(1) | |
|
Ax = the series |
Px |
Px |
Px |
|
Px |
Px |
=1 |
...(2) | |
|
1 = the series |
d |
d |
d |
|
d |
d |
+1 |
.(3) | |
|
1 = the series |
ax 1 |
ax 1 |
ax 1 |
|
ax 1 |
ax 1 |
|
... (4) from (1) | |
|
Ax = the series |
ax 1 —d |
ax 1—d |
ax 1—d |
|
ax 1—d |
az 1—d |
= 1 |
... (5) from (3) and (4) | |
|
Ax=the series |
dAx |
dAx |
dAx |
|
dAx |
dAx |
+Ax |
... (6) from (3) | |
|
The series |
Px —dAx |
Px —dAx |
Px —dAx |
.. |
Px —dAx |
Px —dAx |
=Ax |
... (7) from (2) and (6) | |
|
The series |
dAx |
dAx |
dAx |
|
dAx |
dAx |
=1—Ax |
... (8) from (2) and (7) | |
is a definition of ax
is a definition of Ax and Px and shows that Ax =Px • ax
is a statement of a loan of 1 for the life of (x)
is a statement that ax X ax 1 =1
is a statement which when compared with (2) shows that Px =a1 —d
is a statement of a loan of Ax for the life of (x)
is a statement which when compared with (2) shows that Px • Ax =Px —dAx
shows that dAx =Px(1—Ax)
| Previous | An Introduction to the Theory of Life Contingencies (1931) | Next |