CONCLVSION 1. General Remarks.—The foregoing pages illustrate the standard biometric and contingent functions, their relationships, and their properties in so far as these relate to human life. It may often happen that the actuary has to deal with more compli- cated functions than those considered, but the general principles involved are the same and rest, at bottom, upon the theory of probability and upon the theory of compound interest. At the same time, the actuary may often have occasion to use formulas depending upon the theory of large numbers in its relation to the theory of probability and which do not rest upon so elementary a basis as do those thus far considered. A few of these formulas are added here, more by way of illustrating the desirability of a knowledge of the calculus than with a view to any detailed study of them.1 Consider, for example, Exercise 3, Sec. 19. It is clear that the insurance company will expect to pay, at the end of the first year, 100 of the 5000 policies issued at age a:. The expression for the probability that it will pay exactly 100 of these is 5oooC'loo(0.02)lo°(0.98)4900, but no one would dream of attempting to compute, by direct processes, the numerical value of this expression. 2. Stirling's Formula.—When an expression, like the above, involves factorials of large numbers, Stirling's formula will yield a sufficiently accurate result. This formula may be written , iim wy^ n. „ = „ —^—, where e = 2.71828 . . . and v = 3.1416. The formula will serve for comparatively small numbers. Thus, with a seven-place table of logarithms, the formula gives 1 Cf. YOUNG, T. E., "Insurance," Pitman & Sons, where the value of such formulas is illustrated and emphasized. 106 §3] CONCLUSION 107 20! = 2,422,786,000,000,000,000 with an error of less than one-half of 1 per cent. The approxi- mation is closer the larger the value of n. 3. Probability of Expectation.—With Stirling's formula, the probability of expectation in re trials, namely, p = nC'nrCpMg)'". reduces readily to the simple formula
where
1 h <« p = ,——— = -7=' (1) v2n7rpq VT \/2npq
is a number called the modulus. Thus, in the example just considered
h =
11
\/2 X 5000 X 0.02 X 0.98I4
and the probability of paying exactly 100 of the 5000 policies at the end of the first year is thus „ =___= a56419 = 0.0403. 14Vr 14 4. Deviation.—The probability that an event will occur exactly as many times as expected in a large number of trials is small and a deviation on one side or the other is quite likely to occur. If r represents the number of times an event is expected to occur in n trials and ( represents the number of times the event actually happens, then the number x = ( -r is called the deviation. The deviation is positive or negative according to (> r or « r. Thus, if there are exactly ninety deaths among the 5000 policyholders during the first year, the deviation is -10. 108 MATHEMATICS OF LIFE INSURANCE [§5 5. Probability of Deviation.—The probability y that a devia- tion x will occur in n trials (n being a large number) is given with sufficient accuracy by the Gaussian error law, y^J-e-^', (1) VTT h being the modulus defined in Sec. 3. The graph of Eq. (1) is a normal probability curve. The probability of paying exactly ninety of the 5000 policies the first year is the probability of a deviation of —10 and is, therefore, y = -le-l(>%»« = 0.0242. 14Vir 6. Probability of Deviation Lying between Given Limits.—The probability? that a deviation will occur lying between —a and +a is given by the formula _h_ f+a ^, 2(ha)r W (haV_(haY -, "V^-o6 dx~ -v/^L1 ^^Q-V. 7~y. -J ^ 2(Affl)''y(-lHfea)^ \/i^;w(2r+l)r!' The series in the square brackets converges for all finite values of ha, and quite rapidly for values of ha less than 1. For larger values of ha, the series converges slowly and is not well adapted for computation. But for large values of n, h is always small, and thus Eq. (1) suffices when the limits —a and +a are not far apart. A table of values of P is inserted in most texts on statistics. Glover's tables contain a very full table of values.' The probability that the number of deaths among the 5000 policyholders, during the first year, lies between 90 and 110 (limits included) is found by substituting 1 %4 for ha in the above formula or by making use of a table. By either method, we find the required probability to be P = 0.6876. ' The probability integral Eq. (1) is slightly modified in Glover's tables from the form used here. §7] CONCLUSION 109 7. Probable Deviation.—There are limits —k and +k such that the probability a deviation will fall between them is one-half; that is, such that a deviation is just as likely to fall between them as outside them. With a table of values of P, interpolation gives hk = 0.47694 when P = ^. The number 0.47694 k = ——,— is called the probable deviation. The probable deviation in the 5000 policies is k = 0.47694 X 14 = 7 to the nearest integer. It is an even chance that the number of deaths will fall between 93 and 107 (limits excluded). 8. Mean Deviation.—If each possible deviation, considered as a positive number, is multiplied by the probability of its occur- rence and the products added, the result is called the mean deviation. The mean deviation is indicated by the Greek letter i\ and is given by the formula ^J-f^e-^dx^——— (1) VirJ-m h-V-ir Thus, in the problem we have been considering, r, = -14. = 14 X 0.56419 = 8 VTT to the nearest integer. The probability that a deviation will occur between the limits —77 and +T; is found from Eq. (1) Sec. 6 or from a table of values of P, to be P = 0.5750. 9. Standard Deviation.—The standard deviation is indicated by the Greek letter o- and is given by the formula .T^-^r1"'e-^'xVx^— W \/vJ-» ""
or
<r = ——— (2) AV2 110 MATHEMATICS OF LIFE INSURANCE [§10 The standard deviation for the 5000 policies is = 14 X 0.7071 = 10 to the nearest integer. The probability that a deviation will fall between —a and +a is found when ha- = y^ = 0.7071 and is P = 0.6827. The probability that the number of deaths among the 5000 policyholders during the first year will fall between 90 and 110 (limits excluded) is 0.6827 (cf. Sec. 6). 10. Recapitulation.—For convenience, the above formulas are brought together, » = 0 V = —/= = A X 0.56419 VTT x = k = a47694 y = J-e-^^ = A x 0.44941 h Va- x = n = ,-/= y = -7=e-V' =h X 0.41038 A-VTT VTT ^^ATl 2/=^-^= AX 0.34220