42   Life Contingencies

We can regard ax as an annuity certain for w—x years the successive annual payments being px, 2Px, 3Px . . . decreasing in amount each year over the w—x years, but the total sum to be received would be px+2px+ . . . =ex, which is the total sum to be received under aEx .   So that, while the two annuities-certain,

ax and aexl, have the same total payments amounting to ex, this

total amount is to be become payable sooner under al than under ax.

Therefore aexl must exceed ax.

13. Suppose that we have two lives, one aged x and the other aged y, an annuity payable so long as both of them are alive would be represented by axy, where

An annuity payable so long as there is at least one survivor of the pair (x) and (y) is represented by

axy =ax+ay —axy.

An annuity payable so long as (y) may live after the death of (x) is represented by

ax I Y =ay —axy.

co

h   7   m

axy = vttpxydt=   at~tt'xy(Ilx+t+py+l)dt

0   0

axy =J vt(tpx+tpy —tpxy) dt, 0

co

,/

ax y= 7/(1 —tpx)tpydt 0

co

,h

ax y =   Z'tt pxy • lax+t ay.+t . dt.

0

Also

and or