Approximate Confidence Limits on the Mean of X + Y Where X and Y are Two Tabled Independent Random Variables

1974 ◽  
Vol 69 (347) ◽  
pp. 789 ◽  
Author(s):  
W. G. Howe
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Confidence Limits ◽  
The Mean

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (04) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (4) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

Randomized multihit models and their identification

1996 ◽  
Vol 33 (2) ◽  
pp. 458-471 ◽  
Author(s):  
L. G. Hanin ◽  
L. B. Klebanov ◽  
A. Yu. Yakovlev
Keyword(s):  
Cell Survival ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Independent Random Variables ◽  
Critical Number ◽  
Cell Sensitivity ◽  
Target Model ◽  
Radiation Induced ◽  
Unit Dose ◽  
The Mean

The multihit–one target model induces a stochastic ordering of cell survival with respect to the cell sensitivity characteristics. This property can be used for a description of cell killing effects in heterogeneous populations of cells on the basis of randomized versions of the model. In such versions, either the critical number of lesions or the mean number of hits per unit dose (sensitivity), or both, are assumed to be random. We give some new results specifying conditions under which the randomized multihit models are identifiable, with a focus on the following cases: (1) the critical number of radiation-induced lesions, m, is random; (2) the sensitivity parameter, x, is random given m is known or otherwise; (3) x and m form a pair of independent random variables.

Randomized multihit models and their identification

1996 ◽  
Vol 33 (02) ◽  
pp. 458-471 ◽  
Author(s):  
L. G. Hanin ◽  
L. B. Klebanov ◽  
A. Yu. Yakovlev
Keyword(s):  
Cell Survival ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Independent Random Variables ◽  
Critical Number ◽  
Cell Sensitivity ◽  
Target Model ◽  
Radiation Induced ◽  
Unit Dose ◽  
The Mean

The multihit–one target model induces a stochastic ordering of cell survival with respect to the cell sensitivity characteristics. This property can be used for a description of cell killing effects in heterogeneous populations of cells on the basis of randomized versions of the model. In such versions, either the critical number of lesions or the mean number of hits per unit dose (sensitivity), or both, are assumed to be random. We give some new results specifying conditions under which the randomized multihit models are identifiable, with a focus on the following cases: (1) the critical number of radiation-induced lesions, m, is random; (2) the sensitivity parameter, x, is random given m is known or otherwise; (3) x and m form a pair of independent random variables.

On the duality between the behaviour of sums of independent random variables and the sums of their squares

1978 ◽  
Vol 84 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Absolute Moment ◽  
Mild Conditions ◽  
Standard Normal Variable ◽  
Normal Variable ◽  
Poisson Law ◽  
Standard Normal ◽  
The Mean ◽  
Normal Law

AbstractLet Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

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