Application of probability distributions mixture of safety indicator in risk assessment problems

The main indicator of quality functioning of the aircraft pilotage system is flight safety. Thereby, in accordance with acceptable risk concept, demands with safety indicators are being determined in probability form and circumferential acceptable risk values could be very small order of magnitude 10−6–10−8. Such demands for the precision characteristics of vertical echeloning and automatic landing of civil aircrafts are valid during flights in all weather conditions. At certification of aircrafts these demands should be confirmed. Thereto, the only possible way of demand confirmation with such small risks is statistical mathematical modeling in the wast range of disturbing factors. Mathematical modeling is inseparable part of aircraft piloting system development and it is accepted as legitimate at corresponding verification, which enables the guarantee for adequate aircraft movement modeling. The most complete information about aircraft movement probability characteristics is contained within laws of their probability distributions. In classical mathematical statistics papers there is no investigations of very small and very big probabilities. Therefore for solving the safety problems, connected with needs for investigation “tail” (last) distribution parts, necessary to develop a new methods. One of them is presented into this paper. For problems of statistical modeling of aircraft movement in the wast range of disturbing factors the typical appearance of “breach”—distribution law alteration of analyzed parameters. The cause of the “breach” is realization of unlikely extreme values of disturbed random factors and their mixtures at large quantities of modeling, which reconduct corresponding systems in areas of nonlinearity. It is mentioned that the effect of “breach” is characteristical not only for problems of aircraft piloting but also for the wast range of problems in different practice applied areas—ecology, quality management (Six Sigma methodology) and other.