Abstract
Since the reliability of construction products, by definition, is a probability of failure-free operation, then to assess the correlation characteristics between the parameters of the reliability of construction products that are not observed at the same time, we will use the designations and terms used in the theory of probability and mathematical statistics. Existing statistical methods for assessing the conditional mathematical expectation applied to the construction product reliability are based on the possibility of simultaneous measurement of the value of random variables. Because of this, they get a set of extreme points in the plane of those variables, and then build a regression line estimates based on those parameters. Such methods are obviously unacceptable for the case when the quantities the mentioned random variables are cumulatively unobserved. The solution to this problem will allow us to find estimates (not statistical) of the regression line and a number of other characteristics associated with it for simultaneously unobservable random variables. The algorithm described in the current work illustrates an example of interest in the theory of reliability for finding the extrema of the averaged function over the given initial parameters of the construction product.
Highlights
Since the reliability of construction products, by definition, is a probability of failure-free operation, to assess the correlation characteristics between the parameters of the reliability of construction products that are not observed at the same time, we will use the designations and terms used in the theory of probability and mathematical statistics
It is required to find the absolute minimum ݉(ߎ, ܨ, )ܩand absolute maximum ߎ(ܯ, ܨ, )ܩof the function ܴொ(ߎ, ܨ, )ܩon the set ܵ(ܨ, ܨ(ܮ ∈ )ܩ, ݉ ∩ )ܩഥ(ܨ, [ )ܩ3,5-8] The solution to this problem is of interest in the theory of reliability [5,7], since it can be used to estimate the regression line between simultaneously unobservable random variables
Existing statistical methods for assessing ܫொ are based on the possibility of simultaneous measurement of the value of random variables [ and ߟ [1,3,6,7,8,9,10,11,12,13]. They get a set of extreme points ([1, ߟ1) ([2, ߟ2).... ([n, ߟn) in the plane ([, ߟ), and build a regression line estimates based on those parameters
Summary
Since the reliability of construction products, by definition, is a probability of failure-free operation, to assess the correlation characteristics between the parameters of the reliability of construction products that are not observed at the same time, we will use the designations and terms used in the theory of probability and mathematical statistics. We denote L(F,G) as the set of all conditional distributions Q for which ܫொ( )ݔis a non-decreasing function in x, and ݉ഥ(ܨ, ܳ{ = )ܩ: ∫ ܳ(ݕ/= )ݔ(ܨ݀)ݔ })ݕ(ܩ Assuming all stated above, the problem under consideration can be formulated as follows. An example of such quantities are the moments of failure of a construction product in various modes [6,7,8,9,10,11,12,13] The solution to this problem will allow us to find estimates (not statistical) of the regression line and a number of other characteristics associated with it for simultaneously unobservable random variables. The distributions of F and G can be determined from experimental data, and the monotonicity of ܫொ( )ݔoften follows from physical considerations
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