On Mathematical Statistics and its application to Political Economy and Insurance. By Dr. THEODOR WITTSTEIN, Actuary of the Hanover Life Insurance Company. Translated by T. B. SPRAGUE, M.A.

(Continued from page 189.)

§ 6.

Besides the probable error above considered, there is another quantity of some importance in practice, which may be called the Risk of a departure from the most probable number of survivers. It is obtained by adding together the products of all the possible positive deviations in the number of the survivers from its most probable value, each multiplied into the probability of its occurrence; or by adding together the products of all the possible negative deviations in the number of the survivers from its most probable value, each multiplied into the probability of its occurrence; in this case, however, taken without sign. For if we consider the deviations from the most probable number of survivers, taken without regard to sign, as represented by proportional sums of money, then the quantity just defined evidently expresses either the mathematical expectation of gain of a person who is to receive those sums, or the mathematical expectation of loss of a person who is to pay them. The latter of these two ideas more generally occurs in practice. If we denote this Risk by R, we have from (6)

This formula differs only in its numerical coefficient from that for the probable error, (8); and the observations in §§ 4 and 5 can therefore be easily extended to this quantity.

For instance, we have according to Brune's table, out of 7943 males living at the age 40, 7847 living at the end of a year, with a Risk 3.89; or if an Office, as usually happens, insures a given sum for every death among the 7943 males, it must charge, in addition to the most probable amount of the claims, 96, a sum of 3.89 for the risk, that is, nearly 4 per-cent additional.

This has an important bearing on the calculation of the premiums of life insurance companies. Hitherto, every calculation of life insurance premiums has been based only upon the most

probable number of survivers; and a loading of arbitrary magnitude has been added to the premiums to cover the Risk. If, instead of this, in calculating any premium we were to add to the most probable value of the premium found in the usual way, the corresponding "risk" found by means of the formula (10), we should then have a perfectly intelligible determination of the magnitude of the loading necessary to cover the risk arising from fluctuations in the mortality. Similar remarks apply as regards the value of the policy and the Guarantee Fund that is necessary to cover the risk in question. The details of these calculations I reserve for another opportunity.

Chapter 2.

The probability of living a year is to be deduced from observation.

§ 7.

The probability of living a year is never known à priori, but must be deduced from facts furnished by experience. But since this can never be done with certainty, but only with a greater or less degree of probability, this same degree of probability must also attach to the final result; and the above conclusions consequently require to be modified.

Suppose that observation of L persons of the age has shown us L' of them living at the end of a year. Let y be the unknown probability that a person of the age a will be alive at the end of a year. If y were known, the probability of the above observed fact would be, as in (1),

Consequently, the probability, 2, of the hypothesis y, is

(11)

Ω=

L+1

that is to say,

Q= L' L—L'yL'(1—y)L-L'

where

y

can have all values from 0 to 1.

The value of y for which is a maximum is

(12)

If we denote by yo this most probable value of the probability y, we have

This value is what is commonly, but erroneously, regarded in practice as the true value of the probability sought.

§ 8.

If observations of the above kind have been made for various years, and we desire to deduce a single result from them, then the above investigation must be modified as follows:

Suppose we have observed that

In the 1st year, out of 7 persons of the age x, l' are alive at the end of the year,

Let y be the unknown probability that a person of the age x will be alive at the end of a year. Assuming the hypothesis y, the probabilities of the facts observed in the several years are, by (11),

Consequently, the probability of the concurrence of all these observed facts is equal to the product

we get for the probability of the hypothesis y, the same formula as before, (12); and for the most probable value of y, the formula (13). The result of the investigation therefore is, that we must treat observations made in different successive years as if they were made at the same time upon as many coexistent bodies of persons, and combine these into a single body;—a course which in itself appears reasonable.

Furthermore, this investigation plainly assumes that there is no

reason to suppose that the value of the probability y is different in different years.

§ 9.

If persons have come under observation or gone out of observation (entered or exited) during the year, so as to be under observation for part of a year only, these can be regarded as incomplete observations, and be taken into account as follows:

Suppose we find from observation

A persons living at the beginning of the

C

year,

entering in the course of the year,

exiting

dying

D

and lastly, A'

so that we have

living at the end of the year,

A'=A+B-C-D;

all these persons being

beginning of the year.

supposed to be of the same age x at the Assume now that the entries and exits are distributed uniformly over the year, and let q be the probability that a person of the age x will die in the course of the year.

If a fraction, 0, of the year has elapsed, the probability of dying in the remaining part of the year may be put proportional to that part,* that is, q(1-0). Now, out of A persons alive at the beginning of the year, the most probable number of persons dying in the course of the year is Aq.

Furthermore, out of B persons entering in succession, we have as the most probable number of persons dying after entry

B

2

that is to say, just as many as would die out of living at the beginning of the year.

Lastly, out of C persons exiting in succession, the most probable number of persons dying after exit is

that is to say, just as many as would die out of

*See my paper, "Die Mortalität in Gesellschaften mit successiv eintretenden und ausscheidenden Mitgliedern," in Grunert's Archiv, 39. Theil. 1. Heft.

is the most probable number of persons dying in the course of the year. But with the notation of § 7, this same value is expressed by Lq; so that we must put

and the problem is thus reduced to that of § 7.

Observation. These formulas (17) must evidently not be applied if the supposition we have made above does not hold good, and the entries and exits, instead of being uniformly distributed over the year, fall at fixed times in the year. In such a case, it is clear from the foregoing, that the numbers of persons entering or exiting must be multiplied by the fraction of a year for which they were under observation, and be added to or subtracted from the number at the beginning of the year, in order to get the proper value of L. Thus, for example, the exits in the Berlin Widows' Fund, which furnished the materials for the construction of Brune's table of mortality, took place only at the middle and the end of the year. In that case, therefore, dividing the exits of each year equally between the middle and the end of the year, we must put L-A-C, and consequently L'A' +¿C.

§ 10.

Going back now to § 7, if we put y=yo+u=

whence, with the help of Stirling's formula, if L and L' are large

as the approximate formula for the probability that the value of y will differ from its most probable value by the amount u.