Expectations Of Random Variables Research Articles

A note on the Cauchy–Schwarz inequality for expectations

In this short communication we remark that the well-known Cauchy–Schwarz inequality for expectations of random variables is a consequence of Jensen’s inequality, which does not seem to have appeared previously in the literature.

APPLICATIONS OF MULTIVARIATE AND BIVARIATE ANALYSIS IN SCIENCE AND ENGINEERING

The applicability of multivariate analysis provides the gradual build up for the discussion of multivariate problems, it extends the application procedure to bi-variety distributions. Sample from a multivariate population looks like and how the sample statistics like mean, variance, covariance matrix, correlation matrix can be calculated are explained with the help of a numerical practical illustrations. It has wide applications in many fields like education, medicines, agriculture and Economics. A series of univariate statistical analysis carried out separately for each of the variables. The objectives of scientific investigations for which multivariate methods most naturally implemented in statistical computation methods. Whenever an investigator requires to understand a social or physical phenomena variables to records. Bivariate discrete and continuous random vectors are considered in expansion form of Multifivatirate analysis tabulations. Bivariate distribution which has got an importance of its own will be explained for easy understanding of various methods of analysis in Multivariate and Bivariate application methods. It provides the methods of multivariate techniques, data organization, arrays and basic descriptive statistics, Bivariate continuous distributions methods, expectation of random variables, and marginal mass functions role in Engineering curriculum.

A risk-averse stochastic program for integrated system design and preventive maintenance planning

The failure of high-consequence systems, such as highspeed railways, can result in a series of severe damages. Due to the volatility of real circumstances, stochastic optimization methods are needed to aid decisions on reliability design and maintenance for the high-consequence systems. Traditionally, risk-neutral approaches are used by considering the expectation of random variables as a preference criterion. The risk-neutral approaches may achieve the solutions that are good in the long run but do not control poor results under certain realizations of random variables. From a perspective of risk analysis, such solutions are not acceptable for high-consequence systems. To address this issue, this paper uses the conditional value at risk (CVaR) to more properly account for some of the worst realizations of random future usage scenarios, and then proposes a risk-averse two-stage stochastic programming model to simultaneously determine the numbers of components in each subsystem and preventive maintenance time intervals for the high-consequence systems that are exposed to uncertain future usage stresses. The proposed stochastic programming model can be converted to a nonconvex mixed-integer nonlinear programming (MINLP) model. To solve the model, we derive the analytical properties of the recourse function and the closed form of CVaR and then design a decomposition algorithm. Numerical examples demonstrate the proposed risk-averse stochastic approach and the effectiveness of incorporating the CVaR in modeling. The results show the research problem significantly benefits from the proposed approach. Furthermore, the robustness of the optimal system design and maintenance plan under different profiles of future usage scenarios is addressed.

Strong law of large numbers for upper set-valued and fuzzy-set valued probability

In this paper, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabilities. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.

Reverse logistics network design and planning utilizing conditional value at risk

Nowadays, due to some social, legal, and economical reasons, dealing with reverse supply chain is an unavoidable issue in many industries. Besides, regarding real-world volatile parameters, lead us to use stochastic optimization techniques. In location–allocation type of problems (such as the presented design and planning one), two-stage stochastic optimization techniques are the most appropriate and popular approaches. Nevertheless, traditional two-stage stochastic programming is risk neutral, which considers the expectation of random variables in its objective function. In this paper, a risk-averse two-stage stochastic programming approach is considered in order to design and planning a reverse supply chain network. We specify the conditional value at risk (CVaR) as a risk evaluator, which is a linear, convex, and mathematically well-behaved type of risk measure. We first consider return amounts and prices of second products as two stochastic parameters. Then, the optimum point is achieved in a two-stage stochastic structure regarding a mean-risk (mean-CVaR) objective function. Appropriate numerical examples are designed, and solved in order to compare the classical versus the proposed approach. We comprehensively discuss about the effectiveness of incorporating a risk measure in a two-stage stochastic model. The results prove the capabilities and acceptability of the developed risk-averse approach and the affects of risk parameters in the model behavior.

Some explicit formulas for calculation of conditional mathematical expectations of random variables and their applications

For a couple of linear stochastic differential equations describing the behavior of two random values we solve a problem of finding the mean of one of them while the second one is fixed. Examples from the financial mathematics are presented where the formulas obtained here are used.

Max-Plus Stochastic Control and Risk-Sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamiltonian is used to prove a comparison theorem for viscosity sub- and super-solutions. An example from mathematical finance is given, and an application in nonlinear H-infinity control is sketched.

Characterizations of distributions by moment properties of sums of records and successive maxima

We consider the estimation problem for expectations of random variables that are sums of records and successive maxima. The corresponding upper bounds and characterizations of source distributions are obtained in the case where these bounds are attained, Bibliography: 4 titles.

Analysis of the accuracy of Monte Carlo methods for boundary-value problems using a probabilistic representation

Accuracy problems for statistical simulation algorithms are explored in solving boundary-value problems for elliptic equations in mathematical physics. The algorithms are based on a probabilistic representation of the solutions with the use of appropriate systems of stochastic differential equations (SDEs). The problems in question arise from the necessity to simulate long SDE trajectories and estimate expectations of random variables with strongly asymmetric distributions. Numerical results are presented.

Operating on Functions with Variable Domains

The sum, difference, product and quotient of two functions with different domains are usually defined only on their common domain. This paper extends these definitions so that the sum and other operations are essentially defined anywhere that at least one of the components is defined. This idea is applied to propositions and events, expressed as indicator functions, to define conditional propositions and conditional events as three-valued indicator functions that are undefined when their condition is false. Extended operations of “and”, “or”, “not” and “conditioning” are then defined on these conditional events with variable conditions. The probabilities of the disjunction (or) and of the conjunction (and) of two conditionals are expressed in terms of the conditional probabilities of the component conditionals. In a special case, these are shown to be weighted averages of the component conditional probabilities where the weights are the relative probabilities of the various conditions. Next, conditional random variables are defined to be random variables X whose domain has been restricted by a condition on a second random variable Y. The extended sum, difference, product and conditioning operations on functions are then applied to these conditional random variables. The expectation of a random variable and the conditional expectation of a conditional random variable are recounted. Theorem 1 generalizes the standard result that the conditional expectation of the sum of two conditional random variables with disjoint and exhaustive conditions is a weighted sum of the conditional expectations of the component conditional random variables. Because of the extended operations, the theorem is true for arbitrary conditions. Theorem 2 gives a formula for the expectation of the product of two conditional random variables. After the definition of independence of two random variables is extended to accommodate the extended operations, it is applied to the formula of Theorem 2 to simplify the expectation of a product of conditional random variables. Two examples end the paper. The first concerns a work force of n workers of different output levels and work shifts. The second example involves two radars with overlapping surveillance regions and different detection error rates. One radar's error rate is assumed to be sensitive to fog and the other radar's error rate is assumed to be sensitive to air traffic density. The combined error rate over the combined surveillance region given heavy fog and moderate air traffic is computed.

Logarithmic derivatives of invariant measure for stochastic differential equations in hilbert spaces

We consider a process X solution of a semilinear stochastic evolution equation in a Hilbert space. Assuming that X has an invariant measure ν, we investigate its regularity properties. Logarithmic derivatives of ν in certain directions, are shown to exist under appropriate conditions on the nonlinear term in the equation. A set of directions of differentiability for ν is explicitly described in terms of the coefficients of the equation. In some cases, logarithmic derivatives are represented as conditional expectations of random variables related to an appropriate stationary process. An application to a system of stochastic partial differential equations in one space variable is given

Statistical integrated circuit design

Several statistical design methods that have been developed to minimize the effects of IC manufacturing process disturbances on circuit performance are reviewed. It is shown that statistical design problems can be expressed as optimization problems in which either the objective function or the constraint functions depend on expectations of random variables. The effectiveness of the most recent such method, the boundary integral method is illustrated with several circuit design examples. >

On the simulation of expectations of random variables depending on a stopping time

Birkhoff's pointwise ergodic theorem with the shift operator over [0,1]11yields a new practical method to compute expectations of functionals of stochastic process. Indeed converges to as Nconverges toward infinity, almost surely. By numerical simulations we will explain the efficiency of this method especially when compared to the classic Monte-Carlo one. It will furthermore be proven that under suitable assumptions a. central limit theorem holds. These assumptions are satisfied in most encountered practical problems. It will precisely be fulfilled when with a stopping time T having a moment of order p, p > 2.Moreover, under this assumptions a “weak” law of iterated logarithm applies. Such that: Numerical simulations were processed.

The Teacher's Corner: Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity

The Teacher's Corner: Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity

Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity

Inequalities of Expectations of Random Variables Derived by Monotonicity or Convexity