M. Bp]ETON AND K. Pearson 71 
Since A is 44 years of age, he is = 28"748 years before the probable age at 
death of his class. His expectation of life is accordingly 28"748 + where 
1 
V2 
77 •' 
1 
v27r J +x, 
1 - - L r e-""'''^-^""' dx 
V27r^ +x, 
Calculating this from the tables in the usual way we find : 
Xj = -854 years. 
Thus while A gains 6"414 years from his ancestry he only gains "854 years 
from the fact that he has already survived 44 years. His probable duration of 
life is 73-60 years. Ogle's table would give a man of 44, 22'7 years' expectation of 
life ; fi'om our data A has 29'6 years. The difference of 7 years is substantially 
due to A's good ancestr3^ 
Illustration (ii). — A man of 50 and his wife of 44 years of age have had two 
children, one of whom, a daughter, died at 8, and the other, a son, at 12. The 
man wishes to provide an annuity for his wife, if she survives him. What is her 
expectation of life as a widow ? By Equation (15) of Table B the man belongs to 
a group of men who die at the mean age of 65"7l7 years, with a standard 
deviation of 15 2830. By Equation (25) the woman belongs to a group of women 
who die at the mean age of 61'004 years, witii a standard deviation of 17"9290. 
An erroneous solution of this problem might be obtained in the following manner : 
the man will most probably live 15'717 years, if he died at the most probable age 
of death ; his widow if she lived to her most probable age of death would be 
59"717 years old when he died and have r287 years still to live. This is very far 
indeed from her expectation of widowhood. Sucli a solution fails because the 
probable ages of death of certain arrays of men and women do not determine the 
expectations of life of men and women who have already lived to certain definite 
ages. Working out as in Illustration (i) the expectations of life of the man and 
woman, we find them to be 19'953 and 22'510 years respectively*. Would it be 
correct to say then that the expectation of widowhood is 2"557 years ? This 
again is incorrect ; the years of widowhood are in each case correlated with the 
age at which the husband dies, and we cannot find the mean value of these years 
from the mean ages of death of husband and wife. 
The problem is considerably more complex and we must proceed as follows : 
Let Xi be tiie number of years before or after 65 717 years at which the husband 
* Ogle's life table gives 18-93 and 24-72 respectively. Our man and woman, however, do not 
represent the general population. They have been father and mother and have lost two children. 
