40 Mr. C. J. Hargreavc on the Valuation of Life Contingencies 



following scries of values may be readily deduced, the S imply- 

 ing the sum of all the instances of the form placed under it, so 

 that each expression is a symmetrical function of a^, a^ . . a„. 



<^(l) = {l-fl,)(l-flr2) . . . (l-flJ.No. of terms, 1. 



— <|)'(l) = 2«i(l — flfa) (1 — flj . No. of terms, 71. 



^(j)"{l) = l,a]a^(l-'a^) .... (l--«„) .No. of terms, n — «— . 



5-<^"'(l)=Sflia2«3(l— «4) . . . (1 — «„) .No. of terms, n 



n-1 71-2 



"1 ^^]_ ^__7i_l_'l 



+ — - — <l)(p^{l)=^a^ac^aQ..ap{l—ap+i),.(l—a„) . No. of terms, ti —k—" — - — 



1 



<^('*~^)(l)=S«i«2^3..««-i(l— «n).No. of terms, n. 



-2.3.. (71-1) 

 - 1 



<^(")(l) = flia2flr3 . . . fln • No. of terms, 1. 



The sign is positive when p is even, and negative when p is odd. 

 A single term of the pth expression contains p factors of the 

 form «, and n —p factors of the form ^ —a; and the expression 

 itself is the sum of eveiy term which can be so constructed. 



Let there be n persons whose names are Aj, Ag . . A„ ; and let 

 Gp denote the probability that A^ will be living at any given future 

 date. Then by what precedes, we see that the probability that 

 at the given date there will be exactly p out of the set living, 

 (implying that exactly n—p in number are dead), is 



±^-3L^0<.)(l). 



Thus we have — 



Probability that all are dead=l — Sj H-Sg— ^3+ . . . , 

 Probability that exactly one is living = ^^ — 2^2 + ^^s ~ 4!54 + • • • > * 

 Probability that exactly two are living = ^3 — 3.93 4- 654 — 1 Os^ + . . ; 

 and so on until we arrive at the probability that exactly one is 

 dead, which is Sn-x—nSnj and finally the probability that all 

 are living, which is 5„. 



In like manner, we observe that the expression 



.^(i)-<^'(i)+l,^"(i)- . . . ± 2:^'^"''(i) 



represents the probability that p persons or less (including none) 

 are living at the given date ; and, omitting the first tenn, we 

 have the probabiHty that p persons or less are living, excluding 



