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Expanding the second factor by the Binomial Theorem, it becomes
the other terms containing powers of l in the denominator, and
or finally, since we consider k to be small in comparison with 1,
It may be noticed, that it hence follows, since the extreme values of which k is capable extend from – oo to +00, that
Pdk=1. Observation. The following numerical examples will enable the reader to judge of the accuracy of Stirling's formula applied above.
From this it is seen that even for values of n that are not large, the error of the five-figure logarithm affects only such part
of it as we should consider of doubtful accuracy on account of
uncertainty of our data. But it would be a wholly needless attempt at exactness to use more than 5 places of decimals in calculations of this kind.
$ 3. The probability that the number living at the end of a year will lie between Ip+k and lp-k is by (6)
If this probability is to be equal to }, and o is the value of k for which this happens, we have, as is well known,*
h2 whence, by means of (4), o=6745 vip(1-P)
· (8) and this is the probable error to be expected in the observed number l'; or, in other words, it is an even bet that the number living at the end of a year will lie between the limits
1 The quantity
which is here = ✓ip(1-P), represents, as
hv2 is well known, the mean error of the observations.
It follows that, the mortality being the same, the probable error in the number living at the end of a year is proportional to the square root of the number observed. On the contrary, the ratio ĝ, that is, the ratio that the probable error bears to the number observed, is inversely proportional to the square root of the number observed, and can therefore be made as small as we choose, by taking that number sufficiently large. These conclusions hold also for the mean error, and for every other characteristic
The probable error in the number living at the end of a year is evidently identical with the probable error in the number dying within the year.
If lo, the most probable number living at the end of a year, is known beforehand, as is the case in calculations based upon a given mortality table (see the end of § 1), the equation (8) can be put in the form 7.
* See Gauss's Theoria combinationis observationum. The coefficient .6745 is more exactly •6744897, and its five-figure logarithm is 1.82898.
i that is, it is an even chance that the number living at the end of a year will lie between the limits
For instance, according to Brune's table, out of 7943 males of the age 40, 7817 are alive at the end of a year. If we put these numbers respectively for 1 and lo in the above formula, we get o=6:57; and it is an even chance that if 7943 men of the age
40 are under observation, the number living at the end of a year will lie between 7840:43 and 7853:57.
Observation. If the probability (7) is to have the values :9, •99, .999, :9999, the coefficient :6745 in (8) and (9) is replaced by 1:6449, 2:5758, 3.2918, 3.8906, respectively. (See Gauss, Theoria comb. obs.) Consequently the odds are 9 to 1 that the number living at the end of a year will lie between the limits
Ip+1.6149 Vlp (1-P),
In other words, it is to be expected in a large number of observations that, in every 10 cases in which I such persons are observed, the number living at the end of a year will only once fall outside these limits. Similarly in the other cases.
$ 5. The foregoing calculation is not confined to a single year, but can be at once applied to any term of years for which the value of p is given. In particular, starting from the number living at any age we choose, in a given table of mortality, we can immediately calculate the probable error in the number living at any higher age, by substituting for lo in formula (9) the number living at that age according to the table.
For instance, taking Brune's table for males, in which 7943 is the number living at the age 40, we get the following probable errors in the number living at various ages.
The probable error in the number living attains its maximum
7 value when lo=
that is, for the probable duration of life (in the above table for the age of 66, for which o=30); and beyond that point diminishes down to 0. On the contrary, the ratio
lo or the ratio that the probable error in the number living bears to that number, continually increases.
(To be continued.)
The Law of Life Insurance in France as affected by a recent decision
of the Supreme Court of Judicature. By M. LEON DE MONT
LUC, A.I.A., Avocat à la Cour de Paris. NEVER
was a more complete change suddenly brought about in the laws of a nation by legislative enactment than that which has taken place this year in France in the law of life insurance, in consequence of one single decision of the Supreme Court of Judicature. Up to the present time the construction given to the contract of life insurance in this country has been quite different from what it is in England. As there is no provision of written law that relates to life insurance, it being not even so much as mentioned in the Civil or Commercial Codes, people thought themselves justified in governing it by laws and rules of their
For instance, although it is a principle of law common to both English and French jurisprudence (we may add, to the law of all legislating nations from time immemorial) that choses in action shall necessarily devolve upon our legal representatives after our death, it has hitherto been decided almost universally by French tribunals that an exception was to be made in favour of life insurance policies. By the advocates of that doctrine, the right in the sum assured was thought never to have vested in the person effecting the policy, and the assurance monies were said to be transferred directly, i. e., omisso medio, from the assurer to the party entitled to receive the sum assured; and that sum, accordingly, would not be liable to succession duty.
- To us that doctrine always appeared quite untenable; we could not regard the assurance monies as standing on a different footing from any other portion of the assets a person may be possessed of, and though we were not actually confident of the ultimate success of our opinion, we set forth, in a book written in 1867, the reasons that made us believe that the time would arrive when the decisions of so many Courts, conformable to the doctrine of most people versed either in the law or the practice of life insurance, would be over-ruled some day by the Supreme Court. We triumphed, at last, beyond our most sanguine expectations : the outlawry of Life insurance is now no more; it will henceforward be subject, as any other contract, to the operation of the written law. It may be true that the interest of the public exchequer, which was a loser in the positions maintained by our adversaries, contributed a little in turning the scale of victory; but, be that as it may, the construction of life insurance is now restored to sounder principles. Contrary to the doctrine of the Courts of Caen, Lyon, Colmar, Rouen, Paris, it has been held by the Court of Cassation (Chambre Civile) that the assurance monies form part of the estate of the deceased, like any other description of property. It was an action brought by the Stamp Officers (L'Enregistrement v. Krieg) for the payment of an ad valorem duty upon a sum of 20,000 francs insured by one M. Krieg for the benefit of his legal representatives, and made payable within four months after his death. The policy had been disposed of by M. Krieg in his will, but the legatee not having accepted it, the legal representative of the deceased became entitled to its benefit. The tribunal of Saverne decided for the defendant (21st May 1869), upon the ground that the benefit of a life assurance policy was not deemed to confer a succession; but its decision has been annulled by the above mentioned judgment of the Court of Cassation, which is to the following effect :
“Whereas it appeared that Krieg insured upon his life at the Caisse Générale des Familles, in consideration of an annual premium of 420 francs, the sum of 20,000 francs, made payable to his representatives within four months after his death;