There are two main approaches to the study of mortality laws of the population.
The first approach is expressed in an effort to find a unified mathematical formula of the "death law", to represent the probability of dying as a continuous value, the values of which can be calculated at any time of a person's life, thus specifying the tariff. The drawback of this approach is some averaging, the smoothening of the laws of mortality.
The second approach is to build the mortality tables. It allows you to take into account mortality tables averaged for a given age, but not smoothed out the probability of dying. Its drawback is the lack of merit of analytic smoothing.
In actuarial calculations, the second approach is often used, supplementing it with some assumptions about the nature of mortality between individual years of life.
Many authors have tried to express the dependence of mortality on age by one general formula and give an analytical law of mortality. These attempts, however, were unsuccessful due to the extreme complexity and multiplicity of factors affecting mortality, many of which so far are not even quantifiable.
One of the first attempts to obtain an analytic expression for s ( x ) belongs to de Moivre (1725). He suggested a linear character of the decrease in the survival function:
and for the death rate:
The mortality density is constant: F (x) = 1/ω.
Of course, further research revealed that this hypothesis contradicts the experimental data.
In 1825, the English scientist Gompertz (Sotret) managed to find a more suitable general formula that was later modified by Makeham ( Makeham ), expressing the dependence between mortality and age - a formula that gives for middle ages results close to those obtained from observations. According to this law, the logarithms of probabilities to live the next year of life follow the law of geometric progression. Formula
Gompertz - Meikhem, although it is an approximate law, is widely used in modern practice to equalize the mortality tables.
First Gompertz proposed an exponential formula for the force of mortality:
Then in 1860, Makeham along with a factor associated with the "natural" mortality В <С х, took into account the mortality from external causes not related to aging (for example, death from an accident, etc.) (constant A) and specified the formula Gompertz in the form:
In 1869, he proposed one more linear term, and then the law of mortality derived by him took the following form:
In 1931, Peck (or Perke) proposed the ratio
Widely distributed in the analysis of data such as the "lifetime" received the formula Weibull :
However, the universal formula of mortality, which gives a good approximation for the entire range of ages, does not yet exist. Different laws give different degrees of accuracy for different age groups. Therefore, in practice, use piece combinations of these expressions.
Experts believe that the development of a new drug will increase the expected average life expectancy by 4 years. It is assumed that mortality (before drug development and after) is described by de Moivre's law. Determine how the age limit ω will change.
To solve the problem, we introduce the notation. Let ω be the age limit before the development of the medicinal product; ω 'is the age limit after the development of the drug.
Since mortality is described by the de Moivre law, the residual life time r (x) is uniformly distributed on the segment (0, ω - x ). Consequently, the expected life expectancy before the development of the drug according to the formula (7.52):
Then the life expectancy after the development of the drug:
By the condition of the problem
Response: : The age limit will increase by 8 years.
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