Physically realizable algorithms for the localization of random pulse-point sources

In this paper, we describe algorithms for the optimal search for pulsed-point sources, and the information on their distribution is limited to single-mode functions with a stepped probability distribution density, which makes it possible to physically implement the algorithms.

Polynomials and number sets associated with the probability distribution of the hyperbolic cosine type for even values of the parameter

The polynomials used in the formation of the probability distribution density function of the hyperbolic cosine type are investigated. Earlier, on the basis of a hyperbolic cosine distribution, the author obtained numerical sets, among which not only new ones, but also, for example, the triangle of Stirling numbers, the triangle of the coefficients of Bessel polynomials, sequences of coefficients in the expansion of various functions, etc. In this paper, depending on the natural parameter m and the real distribution parameter β , a new class of polynomials is obtained. For even and odd m , the polynomials are constructed using similar, but different formulas. The article presents polynomials for even values m . Structurally, polynomials consist of quadratic factors. The coefficients of the polynomials, ordered by m , form numerical triangles depending on β . Some relations are found between the coefficients. From the numerical triangles, a set of numerical sequences is obtained, which for integers β are integers. Also, polynomials with respect to x turn out to be polynomials with respect to β . With this interpretation the variable acts as a parameter. New numerical triangles and sequences for different x were found. The overwhelming majority of the obtained numerical sequences are new. The class of polynomials arising from problems of probability theory indicates the possibility of applying the results.

Methods of Estimating the Form of the Probability Distribution Density in Tasks of Processing Measurement Results

Issues associated with methods for estimating the shape of the probability distribution density curve are analyzed in order to classify them when processing measurement results. For example, such nonparametric methods as the method of histograms and frequency polygon, as well as the method of classification of distributions, are considered. It is shown that the values of the anticurtosis and entropy coefficient can be taken as independent features of the form of symmetric distributions. For probability distribution densities that have a one-sided character, such as multiplicative noise, a skewness coefficient should be added to the parameters to consider. Recurrent procedures for obtaining current estimates of numerical characteristics of analyzed random processes are given. The results of processing a random process based on recurrent procedures are presented. It is shown that when the number of samples increases, the estimates obtained by using recurrent and non-recurrent procedures converge. The scattering of estimates of probability distribution density parameters, such as variance, relative mean square error, and entropy error, is determined.

MULTICHANNEL CONVERTION OF RANDOM DATA WITH THE PAIR-ELEMENTS OF ORDERED SAMPLES

The concept of multichannel parallel converting of probability density function (pdf) of random data was previously used for single-element pdf-converters. In development of this concept, here we investigate converting properties of spdf-converters channels formed by the sum of the ​​pairs of ordered sample elements (order statistics). The characteristics of the conversion results as dependencies on the size of the samples and the displacement of the channels relative to the median of the samples were obtained for data with a uniform distribution density. Also where excluded the areas of mutual dependence of the density functions of the summed elements, which further where normalized together with approximating them functions. Despite the apparent structural differences, the goal of this study still was to determine the closeness of the converted data with some standard functions of the probability distribution density, in particular, with the normal distribution law. As before, the estimates of the closeness of the spdf-converter channels were obtained using the chi-square criteria. The results of the research were used to determine the size and location of the statistical closeness windows, and to construct spdf-converters statistical model. References 20, figures 14.