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Probability Theory
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
Remark 4.11 (Conditional expectation) From definition 4.12 the conditional probability distribution Q(.|X = x) of Y given X = x can be used to define the conditional expectation of Y|X = x as a function of x by E(Y|X=x)=∫YyQ(dy|X=x),
Evaluation of Fade Mitigation Techniques
Published in Athanasios G. Kanatas, Athanasios D. Panagopoulos, Radio Wave Propagation and Channel Modeling for Earth–Space Systems, 2017
This value is also a random variable with zero-mean value since the conditional expectation is a conditional estimator and finally, its variance can be obtained E[e2]=(amu)2e(σau)2(e(σau)2−e(σau)2ρndu2⋅e−βut)
Nonlinear Filtering Based on Characteristic Functions
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
We also note that the covariance equation (Equation 9.24) is the same as that given in Lemma 1.2 from Chapter 6 of McGarty [25] except for the term that depends on the measurement residual. Equations 9.25 and 9.26 are based on the Kushner–Stratonovich equation (Chapter 8) for the propagation of the conditional pdf given the measurements. In these equations Ec denotes the conditional expectation which is to be evaluated with respect to the conditional pdf in contradistinction to the evaluation of E in Equations 9.21 to 9.23 with respect to the unconditional joint probability distribution of x, z and x^.
Dissipativity, inverse optimal control, and stability margins for nonlinear discrete-time stochastic feedback regulators
Published in International Journal of Control, 2023
Wassim M. Haddad, Manuel Lanchares
Given , denotes the set and so on. Given and , we say x is nonzero on if . Furthermore, given and a σ-algebra , and expectation of the random variable x and the conditional expectation of x given , under the measure . For simplicity of exposition, we omit the symbol in denoting the expectation and conditional expectation whenever the choice of measure is clear from the context. Specifically, we denote the expectation with respect to the probability space by , and similarly for the conditional expectation. We will also find it convenient to write to mean whenever has the joint cumulative distribution function G.
Robust static output feedback control for hidden Markov jump linear systems
Published in International Journal of Systems Science, 2022
A. M. de Oliveira, O. L. V. Costa, G. W. Gabriel
On a probability space equipped with a filtration , is defined as the expected value operator, as the conditional expectation, and the space of all discrete-time signals -adapted processes taking values in such that is represented by (or simply ). The σ-field generated by the random variables is denoted by .
Finite horizon stochastic H 2/H ∞ control with discrete and distributed delays
Published in International Journal of Control, 2021
Let be a standard 1-dimensional Brownian motion on a complete probability space . The information structure is given by a filtration , which is generated by and augmented by all the P-null sets. We assume the dimension of Brownian motion d=1 just for the simplicity of notations. In fact, all the conclusions in this paper still hold true for the case that the dimension of Brownian motion . Define be the conditional expectation of random variable X with respect to . For any , Euclidean spaces or sets of matrices K, the following notations will be used throughout the paper: