Variance rates

The above discussion is used to derive a model of the behaviom of asset prices sometimes referred to as geometric Brownian motion. The dynamics of the asset price X are represented by the ltd process shown in Equation (2.18), where there is a drift rate of a and a variance rate of b X, ... 

Index Precision Critical value Degree Relative error a Relevant degree e Ave Variance rate c. Small Prob errot Po... 

The behavior of underlying asset prices follows a geometric Brownian motion, or Weiner process, with a variance rate proportional to the square root of the price. This is stated formally in (8.11). 

Once numerical estimates of the weight of a trajectory and its variance (2cr ) are known we are able to use sampled trajectories to compute observables of interest. One such quantity on which this section is focused is the rate of transitions between two states in the system. We examine the transition between a domain A and a domain B, where the A domain is characterized by an inverse temperature - (3. The weight of an individual trajectory which is initiated at the A domain and of a total time length - NAt is therefore... 

With the addition of increasing amounts of electrolyte this variance decreases and an approximate linear relationship between internal and external pH exists in a 1 Af electrolyte solution. The cell-0 concentration is dependent on the internal pH, and the rate of reaction of a fiber-reactive dye is a function of cell-0 (6,16). Thus the higher the concentration of cell-0 the more rapid the reaction and the greater the number of potential dye fixation sites. 

Let us consider the overhead-cost data for Table 9-39 with 10 million kg per month as the standard production rate. The static budgeted overhead is then 150,000 per month, or 1.5 cents per kilogram. We assume that the actual overhead is 186,000 for a month in which 12 milhon kg was produced. Then, the static budgeted overhead cost would be 12 million(I.5), or 180,000 per month. Therefore, the variance is 186,000 — 180,000 = -t- 6000, which is unfavorable because 6000 more was spent than was anticipated. 

From Table 9-39 we find that the flexible budgeted overhead cost for a produc tion rate of 12 million kg per month is 190,000. The corresponding variance is 186,000 minus 190,000, or — 4,000, which is favorable Because 4,000 less was spent than was anticipated. Thus, the use of flexible budgeting makes this particular performance look better without changing either the production rate or a single cost of the planned budget. 

From Eq. (9-219) we calculate the direct-labor-cost variance as 3401 per period. This variance is unfavorable. However, by using the relations of Eq. (9-220), we calculate a direct pay, or wage-rate, variance of 0(cl — Cl) = 6509(8.45 — 8.00) = 2929 per period. This direct pay variance is unfavorable. Likewise, we calculate the direct-labor-efficiency variance as Cl(0 — 0 ) = 8.00(6509 — 6450) = 472 per period. This variance is also unfavorable. 

For this example, the adverse direct-labor cost variance of 3401 is due to both a hi er wage rate per hour and a higher number of labor-hours. 

The direct-labor-cost variance can, if necessary, be broken down into a direc t-labor-idle-time variance in addition to the direct-wage-rate and direct-labor-efficiency variances. The direc t-labor-idle-time variance is simply the number of idle labor-hours in the period multiplied by the standard wage rate. This is rarely relevant to the conditions existing in process plants except when maintenance is involved. 

In Eq. (9-223), Cqh — 0Cboh) E known as the budgeted overhead-cost variance, 0(cboh oh). s the overhead-volume variance, and voh(Q Q ) 3.S the overhead-efficiency variance. The last is analogous to the labor-efficiency variance and is the standard overhead rate multiplied by the deviation in time taken to produce a given output. 

Also in Eq. (9-224), Cboh E simply the flexible budgeted overhead cost in dollars per hour for the actual production rate, and the overhead-volume variance 0(cboh oh) is the actual time taken to produce a given output multiplied by the difference between the flexible budgeted overhead cost and the standard overhead cost in dollars per hour. The budgeted overhead-cost variance (Cqh — 0Cboh) is the difference between the actual overhead cost and the actual time (in hours) required to produce the given output multiphed by the flexible budgeted overhead cost (in dollars per hour). 

Various Langmiiir-Hinshelwood mechanisms were assumed. GO and GO2 were assumed to adsorb on one kind of active site, si, and H2 and H2O on another kind, s2. The H2 adsorbed with dissociation and all participants were assumed to be in adsorptive equilibrium. Some 48 possible controlling mechanisms were examined, each with 7 empirical constants. Variance analysis of the experimental data reduced the number to three possibilities. The rate equations of the three reactions are stated for the mechanisms finally adopted, with the constants correlated by the Arrhenius equation. 

On page 4, rates are calculated for the four specified conditions. Variance is calculated in the experimental results and correlation coefficients are used to show that fraction of the variance in the experimental results accounted for by the model. This is over 99%. Finally the experimental error is calculated from the repeated experiments on page 5. 

Surface roughness to process risk FMEA Severity Rating, strength Ultimate tensile strength Uniaxial yield strength Bilateral tolerance Unilateral tolerance Tolerance to process risk Variance Class width... 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

In summary, the rate theory provides the following equations for the variance per unit length (H) for four different columns. 

Equations that quantitatively describe peak dispersion are derived from the rate theory. The equations relate the variance per unit length of the solute concentration... 

Figure 10. Graph of Variance against Flow Rate for Coiled and Serpentine Tubes...

Variances in resin performance and capacities can be expected from normal annual attrition rates of ion-exchange resins. Typical attrition losses that can be expected include (1) Strong cation resin 3 percent per year for three years or 1,000,000 gals/ cu.ft (2) Strong anion resin 25 percent per year for two years or 1,000,000 gals/ cu.ft (3) Weak cation/anion 10 percent per year for two years or 750,000 gals/ cu. ft. A steady falloff of resin-exchange capacity is a matter of concern to the operator and is due to several conditions ... 

A weighted least-squares analysis is used for a better estimate of rate law parameters where the variance is not constant throughout the range of measured variables. If the error in measurement is corrected, then the relative error in the dependent variable will increase as the independent variable increases or decreases. 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

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