The paper examines the concept of „conditional expected value”, which is of great importance in modern finance. The considerations are carried out on a five dimensional random vector with multivariate t-Student distribution. In the first part we construct a distribution of its coordinates in a 2:3 ratio (i.e., the vectors are two-and three-dimensional, respectively) in order to find an effective two-dimensional vector regression function in relation to the three-dimensional vector. To that end, the probability density distribution of the boundary three-dimensional vector is determined (by calculating the appropriate double integral), and then the conditional probability density distribution of two-dimensional vector was used to produce the three-dimensional vector. The second part of the paper discusses the reasoning presented in the first part and then generalises it for a random vector of any size that will remain applicable provided that it is a multi-dimensional random vectors with t-Student distribution. The results (the general form of the regression function) are illustrated with a specific quantitative example that maintains a „hyperplane” regression.
The paper looks at several results from research on the stability of different graph properties. Definitions of Bondy-Chvátal’s closure and stability for simple graphs are first presented, followed by an overview of basic facts on the stability of selected simple graph properties, for which stability has been established exactly. Proofs for theorems concerning a new example are included. Papers in which closure operation or stability of graph properties have been applied are also presented.
The aim of the study is to compare the effect of changes in fertility and mortality on population aging in Poland by provinces. The following hypotheses were verified:(1) the main factor affecting the aging of the population in Poland is the decline of fertility, (2) lengthening of life expectancy of elderly people has a weaker effect, due to the lower rate of decline of mortality in this population, (3) aging of the population in Poland varies depending on the place of residence. To verify these assumptions, considered two types of scenarios. The first one shows the evolution of population structure, individual regions, assuming a constant level of fertility, which are assigned to hypothetical changes in mortality. In the second indicated how changes in population structure, with varying fertility attributed to a constant level of mortality.
The aim of this article is to present some generalisations of actuarial problems which have interesting solutions from a mathematical point of view. The generalisations we propose can be used in practice to calculate net single premiums in some types of life insurance. Solutions of problems are non-trivial and include advanced mathematical analysis techniques and algebra required in applied mathematics. The problems considered here are connected with a statistical model of life insurance (the distribution function of the future lifetime, force of mortality) and mainly the calculation of net single premiums in chosen types of life insurance. It is also presented very interesting example in which a net single premium of a whole life insurance policy for a person at fractional age is calculated (for a person at the age of x + α, where x is an integer number and α is a fraction, 0 ≤ α ≤ 1). There are proven relationships between the net single premium calculated for different assumptions of mortality during the year. Example illustrated values of these premiums for different fractional parts of the age are provided.
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