Probabilities of Death and Survival   25

Making a total of 2lx+lx+1+Ix+2+ . . . =Tx years.

The share of each of the 1, persons in this total is z+px+2Px+3px+ ... to end of table.

This is called the "complete expectation of life" at age x and is

written, ex = d xx = 2 -~ E npx = E ,:px -1.

   n=1   n=0

The "curtate expectation of life", written ex, is the average number of completed years of future life after age x. Since the lives will on the average die in the middle of the year of death,

ex=z+ex•

The probability that (x) will die in the (n+1)th year from

now after completing n full years following age x is n I qx =dx+n . lx

Since the sum of these probabilities for all integral values of n is 1, the mean or expected number of completed years is co   °'

ex= E•.IgxXn=— E ndx+,i

n=1   lx n=

1

= — [dx+1+2dx+2+3dx+a+    I

lx

= 1 [lx   lx       1, as before. -{-1 +l x~-2 +~-3 +

Again, of the lx persons, 1x+1 will survive the first year, lx+2 will survive the second year, and so on, so that there will be 1x+1 completed years lived during the first year,

1x+2 completed years lived during the second year,

lx+3 completed years lived during the third year,

The total number of completed years therefore will be lx+1+lx+2+lx+a+ . . . to end of table, and the average will be ex =Px+2px+3Px+ ...