26   Life Contingencies

Again, consider one of the l,- persons. He may live through the nth year in which case he will have lived through one additional year of life, the probability of his doing so being npx. His curtate

expectancy, therefore, is ex = E npx X 1.

n=1

Also the expectancy of life during the next n years is nex = ex ,—i = E rpx = ex —npx ex+n •

r=1

And the deferred expectancy after n years is nIex=ex — ex, =nt'.xex+n

But ex n = ex —npx ex+n

ex—npx ex+n + 2(1—npx = ex7r + (1—npx)•

9. Consider the curve y =lx+e

Y .5

In the above curve

ex Xlx =the area of the rectangle OL.

=the sum of the areas of all the rectangles OP1, AP2, BP3 . . . under the curve.

But ex =J ,px.dt 0