Mathematical expectation

The mathematical expectation of a discrete random variable X is defined as 

Consider X being the face value of a rolling die. The expectation is (X) = 3.5. 

Note that for simplicity, we will not write the limits of integration from now on. 

Generally for any function g(X) of a discrete random variable, or g(x) of a continuous random variable, the expectation of the function can be calculated as 

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The symbol E[faX)] or E[] is referred to as the mathematical expectation of the function fa The use of this term stems from the early applications of the theory to games of chance, where it was used to denote the average amount a gambler could expect to win. The basic rules for manipulating the expectation symbol E are direct consequences of its definition, and are stated below for easy reference. 

The notions of random variable and mathematical expectation also cany over to the multidimensional case. A function of n + m-real variables is called a random variable when it is used to generate a new time function Z(t) from the time functions X(t) and Y(t) by means of the equation... 

Under very general conditions, it is true that the empirical mean given by Eq. (5), converges to the mathematical expectation (3), as / - oo (Vapnik, 1982). On the other hand, it is clear that... 

AEr occurs in the denominator AEr always summed together with Er. Together with point 2, this means that the denominator is never zero as long as the reference energy Et is non-zero. Therefore all terms to be included in the computation are finite. Again we repeat our reminder that the results of these computations are mathematical expectations, in a real measurement situation denominators of zero can be expected to occur when Er is less than approximately five. 

If we suppose that two arrays of time delays Mi and M2 correspond to the normal distribution law and they are not homogeneous then the attack is carried out using intermediate computers [12], Therefore, it is necessary to fulfill the appropriate analysis in order to determine the attack source status. For this purpose it is necessary to calculate mathematical expectations xv for an array Mi(f) and yv for an array M2(0, dispersions Dvx, Dvy and mean-square errors crvx and [Pg.197]

The mathematical expectation, dispersion and standard deviation are as follows ... 

Mathematical expectation—The average value obtained if the random variable is measured exceedingly many times. 

The mathematical expectation of the activation barrier was calculated to be AE = 1.7 eV. It is significantly less thmi the barrier AE = 2.5 eV for the diffusion of O atom in Si bulk crystal. The calculation of prefactor (Do) was carried out as in [6] to be Do=4.5-10" cm S . The analysis of hydrogen diffusion in SiNT has shown, that, unlike O2 molecule, the H2 molecule is stable in SiNT. For H2 there exist several conformations, differing orientation of the molecular axis with respect to the SiNT axis. The calculated barrier is close to the diffusion barrier of H2 in bulk Si crystal. The calculated prefactor for diffusion of H2 molecule in SiNTisDo=3.9-10- cm -s. ... 

Trueness of measurements is defined as closeness of agreement between the average value obtained from a large series of results of measurements and a true value. The difference between the average value (strictly, the mathematical expectation) and the true value is the bias, which is expressed numerically and so is inversely related to the trueness. Trueness in itself is a qualitative term that can be expressed as. 

The characteristic function of the random variable X is the mathematical expectation where a is an auxiliary variable and i is the imaginary unit. For... 

Both mathematical expectation and the dispersion of incidental value are characterized by the most important distribution features — its posture and the order of incoherence. The dissipation interval of the incidental value around its mathematical expectation characterizes the average quadratic divergence (1 ,). The Bessel function for the estimation of average quadratic error of the observation result can be represented as follows ... 

It should be mentioned that actually represents the distribution of the mathematical expected value of the particles, and (as in kinetic theory) higher-order distribution functions are required to rigorously determine the system evolution-see Ramkrishna, Borwanker, and Shah [115, 116, 117]. However, Eq. 12.6-1 or 2 seem to be adequate for a variety of problems. 

Industrial engineers frequently use simulation experiments to compare the performance of alternative systems and, ideally, to optimize system performance. When a system is modeled as a stochastic process, the objective is often to optimize expected performance, where expected means the mathematical expectation of a random variable. This section describes methods for optimization via simulation, using the problem of selecting the inventory policy that minimizes long-run expected cost per period as an illustration. 

However, if don t use of this detail information, but to be concentrated only on the value z, it can be find without taking into account of the binomial distribution law. Really, since the / is an average upon the all trajectories probability to discover the occupied cell, the mathematical expectation of the number of occupied cells at 2independent tests will be equal to ld- )p. Correspondingly, the mathematical expectation of the number of empty cells under the same 2independent tests will be equal to (2d- ) -p). 

The probabilistic analysis of the SARW trajectories determines z as the mathematical expectation of the number of free among 2d-l neighboring cells via average upon the all SARW trajectories probability p to discover the occupied cell. It leads to the expression ... 

In a view of the size distribution law it is not difficult to calculate the average value (mathematical expectation M) of the force acting on the spherical particles with their volume V = ttD I6 according to the formula... 

Dong, W., Zhang, W., Chen, G., Liu, J., Radiation effects on the immiscible polymer blend of nylonlOlO and high-impact polystyrene (HIPS) I Gel/dose curves, mathematical expectation theorem and thermal behaviour. Radiation Physics and Chemistry 2000, 57(1), 27-35. 

Risk Mathematically, expected loss the probability of an accident multiplied by the quantified consequence of the accident (SSDC) an expression of the possibility of a mishap in terms of hazard severity and hazard probability (MIL-STD-882) note Hazard exposure is sometimes included (AFR 800-16) as defined in NHB 5300.4(10-2), The chance (qualitative) of loss of personnel capability, loss of system, or damage to or loss of equipment or property (NSTS 22254) a measure of both the probability and the consequence of all hazards of an activity or condition. A subjective evaluation of relative failure potential. In insurance, a person or thing insured (ASSE). 

Because the probability of a, taking any value between 0 and 90° is equal, O, obeys mathematical equi-partition law in the range of 0-90°. Therefore, a, = Eipii), where a, is the average value of a,- and E ai) is the mathematical expectation of at. According to mathematical equi-partition law, it can be obtained as ... 

The average adhesive force, i.e., the mathematically expected value, is expressed by the formula... 

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