Stochastic variables

One approach to a mathematically well defined performance measure is to interpret the amplitude values of a processed signal as realizations of a stochastic variable x which can take a discrete number of values with probabilities P , n = 1,2,..., N. Briefly motivated in the introduction, then an interesting quality measure is the entropy H x) of the amplitude distribu-... 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

This is a technique developed during World War II for simulating stochastic physical processes, specifically, neutron transport in atomic bomb design. Its name comes from its resemblance to gambling. Each of the random variables in a relationship is represented by a distribution (Section 2.5). A random number generator picks a number from the distribution with a probability proportional to the pdf. After physical weighting the random numbers for each of the stochastic variables, the relationship is calculated to find the value of the independent variable (top event if a fault tree) for this particular combination of dependent variables (e.g.. components). 

The incorporation of discreet nucleation events into models for the current density has been reviewed by Scharifker et al. [111]. The current density is found by integrating the current over a large number of nucleation sites whose distribution and growth rates depend on the electrochemical potential field and the substrate properties. The process is non-local because the presence of one nucleus affects the controlling field and influences production or growth of other nuclei. It is deterministic because microscopic variables such as the density of nuclei and their rate of formation are incorporated as parameters rather than stochastic variables. Various approaches have been taken to determine the macroscopic current density to overlapping diffusion fields of distributed nuclei under potentiostatic control. 

Dixon, L. C. W. and L. James. On Stochastic Variable Metric Methods. In Analysis and Optimization of Stochastic Systems. Q. L. R. Jacobs et al. eds. Academic Press, London (1980). 

II with a new chapter (for the second edition) on global optimization methods, such as tabu search, simulated annealing, and genetic algorithms. Only deterministic optimization problems are treated throughout the book because lack of space precludes discussing stochastic variables, constraints, and coefficients. 

The unperturbed Hamiltonian 3 is the same for all systems and is time-independent. The time-dependent perturbation G(t), different for each system, is considered as a stationary stochastic variable. We may, without loss of generality, suppose that the mean value of G(t) over the ensemble is equal to zero. We denote by a,p,y,. . . the eigenstates of supposed to be non-degenerate, and by fix, the corresponding energies. 

The equivalence between the stochastic variables and the At-transformed variables is given by the relation... 

Due to the fact that the measurands are stochastic variables an uncertainty propagation analysis was carried out. An uncertainty analysis can answer two questions (1) the expected accuracy (uncertainty) of the method, that is, the expected uncertainty with respect to the sought quantity (2) the most uncertain (sensitive) measurands. 

The theory behind every measurement method can be generalised by Eq (1) [1]. Some quantity (or quantities, measurands) is measured, which has a specific relationship to the sought quantity. The measurand can be regarded to be a stochastic variable associated with an uncertainty, which implies that the sought quantity is also a random variable. The mathematical relationship depends on the physical model, that is, the model of the physical phenomenon of interest, for example temperature, pressure, and volume flow. The physical model always includes limitations, which implies that the measurement method has restrictions that is, it will only function in a certain measuring range and according to the assumption of the model. 

We consider a system whose state at time t is characterized by the value of the stochastic variable X t) 2 We define the joint probability... 

A random number or stochastic variable is an object X defined by... 

This is called a probability measure. Any other set of numbers f(A) assigned to the subsets is a stochastic variable. In agreement with our program we shall not use this approach, but a more concrete language. 

The set of states and the probability distribution together fully define the stochastic variable, but a number of additional concepts are often used. The average or expectation value of any function /(X) defined on the same state space is... 

The characteristic function of a stochastic variable X whose range / is the set of real numbers or a subset thereof is defined by... 

Remark. A logician might raise the following objection. In section 1 stochastic variables were defined as objects consisting of a range and a probability distribution. Algebraic operations with such objects are therefore also matters of definition rather than to be derived. He is welcome to regard the addition in this section and the transformations in the next one as definitions, provided that he then shows that the properties of these operations that were obvious to us are actually consequences of these definitions. 

Averaging is a different kind of operation since it associates with a stochastic variable a non-stochastic or sure number. Alternatively it may be viewed as a projection in the following way. The set of all stochastic variables contains a subset of variables whose probability density is a delta peak. This subset is isomorphic with the sure numbers of the range and may therefore be identified with them. The operation of taking the average is then a projection of the total space of stochastic variables onto this subset. 

Exercise. In the space of stochastic variables a scalar product may be defined by (17). Prove that with this definition the projection onto the average is a Hermitian operator. 

To each step corresponds a stochastic variable Xj (j = 1,2,..., r) taking the values 1 and — 1 with probability each. The position after r steps is... 

Exercise. Let Xj be an infinite set of independent stochastic variables with identical distributions P(x) and characteristic function G(k). Let r be a random positive integer with distribution pr and probability generating function /(z). Then the sum 7 = Xl+X2 + +Xr is a random variable show that its characteristic function is f G k)). [This distribution of 7 is called a compound distribution in feller i, ch. XII.]... 

Let Xl9X2,...9Xr be a set of r independent stochastic variables, each having the same Gaussian probability density Px(x) with zero average and variance o2. Their sum Y has the probability density... 

Example. It is illuminating to see explicitly how the probability distribution tends to its limit. ) Let X be a stochastic variable that takes the values 0 and 1 with probability each. Let Y be the sum of r such variables. Then Y takes the values... 

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