Expected value weighted average

E Expected value (weighted average) in decision-making G. Hypothesis testing... 

Equilibrium average properties are calculated using a statistical weighting of the probability Pq(r) of Eq. (3) raised to the power of q as required by the generalized statistical mechanics. The so-called q-expectation value is written... 

Let us consider a proposed project in which there is a probability pi that a net present value (NPV)i wih result, a probability p2 that (NPV)2 will result, etc. A weighted average (NPV),, known as the expected value, can then be calculated from... 

The average basal metabolic rate for humans is about 65 keal/h, or 1600 keal/day. Obviously, the rate varies for different people depending ori sex, age, weight, and physical condition. As a rule, the BMR is Jow er for older people than for younger people, is lower for females than for males, and is lower for people in good physical condition than for those who are out of shape and overweight. A BMR substantially above the expected value indicates an unusually rapid metabolism, perhaps caused by a fever or some biochemical abnormality. 

Since the SEC/LALLS technique always yields a weight-average molecular weight (l )y for the slightly polydisperse fraction at V, a small overestimation of the sample Rn is expected (, 1 ). As noted previously (Results) a 1% to decrease in the narrow MWD polystyrene Mp values (Table I) accompanied application of the band-spreading correction ... 

Equations (5.2)—(5.4) and Figs. 5.1-5.3 illustrate the nature of the structural observables obtained from gas-electron diffraction the intensity data provide intemuclear distances which are weighted averages of the expectation values of the individual vibrational molecular states. This presentation clearly illustrates that the temperature-dependent observable distribution averages are conceptually quite different from the singular, nonobservable and temperature independent equilibrium distances, usually denoted r -type distances, obtained from ab initio geometry optimizations. 

The chemical shift and line width observed for each water content were taken as weighted averages of the values in the free and bound states, and from two equations expressing these averages, Fb and Ff, the mole fractions of bound and free Na+ ions, respectively. were extracted. Fb significantly increases as the approximate hydration number that might be expected for a SOs Na pair is approached from the direction of considerable hydration. 

Equation (11.8) reads The average of the expectation values of r — for the various valence AOs of atom I, weighted by the rations of the orbital populations to the total atomic population of atom I equals the inverse of the — / distance. For all their their simplicity, Eqs. (11.7) and (11.8) cannot be tested numerically by direct calculation. The reason is linked to the difficulty of partitioning the total electron density into atomic contributions, in spite of an important conceptual step forward due to Parr [219]. A practical step in the same direction is in the construction of suitable in situ valence atomic orbitals (VAO) from accurate ab initio computations [143], as advocated long ago by Mulliken [220] and discussed by Del Re [221]. As will be seen, such in situ VAOs do provide useful information, but they are of no help in solving the additional problem of assigning suitable populations to the orbitals and of dividing overlap populations into atomic contributions. In view of this situation, we take Eqs. (11.5) and (11.8) as statements whose validity rests on experimental evidence, at least for saturated hydrocarbons. 

In Table IV the heats of formation of several nitroaromatic compounds in the ideal gas state are calculated from the measured heats of formation of the solid and the measured heats of sublimation. In cases where the heat of sublimation is not measured at 298°K there should be a correction for the differences in heat capacities of the solid and ideal gas. The data required to make these corrections are not available but in general it is expected that the corrections will be small and can be neglected. From the heats of formation of each compound in the ideal gas state, a value for the group Cb-N02 (ideal gas) has been derived (Table IV). A weighted-average value (CB-N02(ideal gas)) = 3.0 kcal/mole was used, and a heat of formation was estimated for each compound. In Table IV the difference between observed and estimated heat of formation in the ideal gas state is less than + 3.2 kcal/mole in all cases... 

It is noteworthy that the Tg of the CPS-20/BVPy-45 (70 30, w/w) complex is about 50 °C higher than the weight-average value. This fact allows us to expect a possibility of producing complex blends which are more heat resistant than ordinary miscible blends. 

Figure 17 and Table 4 show the dependence of the rate constant k on the weight average of the degree of polymerization as indicated. This dependence was calculated from Eq. (28c) and assumed to be independent of the temperature in the range 10 to 30 °C of the measurements with the system PS/CHX (cf. Ref.5), p. 2853). The P-dependence of the axial transport rate x, calculated from Eq. (25) for the mean overall volume rate w = 15 cm3/h of CHX at three typical column temperatures is also shown in Fig. 17 and Table 4. Fig. 18 and Table 5 show the dependence of the corresponding rate constants kg for the reversible rediffusion, and kg for the retarded rediffusion of the polymer in the flow-equilibrium, at the column temperature for four typical average degrees of polymerization. These values were calculated from Eqs. (24) and (21), respectively, using ks from Fig. 17. It can be seen that the functions kg(T P) (dashed in Fig. 18) represent asymptotes to the functions k T P) (full lines), as expected.

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