Distribution functions exponential

First Order Stoppage Alone. If stoppage is determined solely by a first order process, such as transfer, the foregoing analysis predicts a nearly exponential distribution function. The polydispersity index must then be very close to 2.00. The same result is obtained for bulk and solution polymerizations dominated by chain transfer. Compartmentalization thus has no major effect on the polydispersity of the polymer produced, as was recognized by Gerrens (11), if the stoppage process is dominated by chain transfer. This contrasts with the significant effects of compartmentalization if bimolecular events dominate termination. 

It can be established by the following reasoning. If n = %, each particle contains at most one free radical. Growing chains in the latex particles can thus either grow or be terminated instantaneously by entrant free radicals. These mutually exclusive kinetic events immediately prescribe the Flory most probable distribution function for the growing chains (12) this is an exponential distribution function with a polydispersity index of 2.00 (13). 

The exponential distribution function on chemisorption activation energy conforms to Howar and Taylor s, Banham and Burt s as well as to Kwan s equations. They were used long ago for the description of small molecule chemisorption kinetics on oxide surfaces [116]. It is suggested that apparent rate constant depends on the coverage heterogeneous surface as follows [117] ... 

For an exponential distribution function, the mean time to failure and the characteristic time T are identical. When / = 1 the Weibull distribution represents the region of the working life. For fS > 1 the failure rate increases with time, as when the product is wearing out... 

The Freundlich isotherm has no definite maximum adsorption or saturation value therefore, when 0 approaches a monolayer, the Freundlich equation necessarily breaks down. For high coverages the exponential distribution function is conveniently replaced by a linear distribution... 

The exponential function. The mathematical forms chosen to this point were those considered possible for tails. We now consider the exponential distribution function that may be relevant to loopy adsorption, at least for high molecular weight polymers. The exponential distribution function must be artificially truncated at the physical limit of the steric layer. This leads to a normalized segment density distribution of the form... 

Exponential distribution function. Smitham and Napper (1979) showed that for an exponential segment density distribution function... 

Figure 4 (a) Exponential Density Function, (b) Exponential Distribution Function, (c) Reliability... 

Reliability engineers often make the assumption that the probability of repair is an exponentially distributed function in which case the "restore rate" is a constant. The lower case Greek letter mu is used to represent restore rate by convention. The equation for restore rate is ... 

The methods for evaluating unknown parameter of the component multistate exponential reliability function in various experimental cases with a special stress on small samples and unfinished investigations are defined and formulae for evaluating the intensities of the component departure from the reliability state subsets in all cases are proposed. The common principle to formulate and to verify the hypotheses about the exponential distribution functions of the lifetimes in the reliability state subsets of the multistate system components by chi-square test is also discussed and summarized in easy steps. 

At the beginning of simulation, it is supposed that all components of the network are in operation. Then it is necessary to generate a random variable, which represents a failure of a component, for each component. Program mainly uses exponential distribution function, /(t) = the sampled state duration... 

One can also derive the exponential distribution function heuristically. The random binary model assumes the number of bubbles, larger than a specific volume, m, decreases with volume in direct proportion to itself. That is, the greater the abundance of bubbles larger than a given size, the stronger the effect of random coalescences that reduce their number. This proportionality can be written as... 

The exponential distribution function. Equation 12 is a solution of Equation 14 with the initial condition that F (Oik) = 1 and using the definition of the density... 

The exponential distribution function F t) red curve) expresses the probability that the "entit/ -looking at it from the present moment 0 - will be dead" by the moment t (i.e., that it will decay somewhere between 0 and t). The blue curve, on the other hand, shows the probability of the complementary event, i.e., that the same "entity" will survive the period (0, t). For t > 0 the curve of the exponential density function is obtained from the blue curve by multiplying the latter with A (However, for t < 0, the density function is zero, not 1, as shown in O Fig. 9.1)... 

With the assumption that g(r,9,b) is spherically symmetric the escape probability depends only on r, and p(r,9,E,T) is given by Equation 30. For g(r,b) various analytical functions have been proposed (see, for example, Casanovas et al., 1988). The exponential distribution function. 

This representation of P(0,0) is a truncated expansion of a general exponential distribution function described in Section 4. The expansion P(d,0) is therefore only valid for small values of the average solute orientation. This condition is usually fulfilled in nmr spectroscopy where only small molecules can be studied as yet. The number of independent motional constants, Cj, is considerably reduced if the solute molecules exhibit some symmetry. 

A quantitative analysis of the relative intensities is only possible on the basis of a very elaborate computer simulation of the spectra [60]. Such an analysis yields an estimation of the orientation distribution function of the solute molecules in the liquid crystal matrix [59]. As will be shown in section 4, it is physically reasonable to assume an exponential distribution function, which (for molecules of Djh" and C2crsymmetry) is given by... 

Some recent work in this area has shown the gamma distribution function to represent the particle hit data better than the exponential distribution function. Transfer coefficients for this penetration analysis are given in Example 6-6. 

The Lambda-Tau is a traditional methodology for system reliability analysis, and this method is suitable for the FTA model. The failure rate of every component required to obey exponential distribution function and the failure rate X and the repair time T of the system obtained by the logical gates of the system s FTA. 

Weibull distribution function is one of the widely used cumulative probability functions for predicting lifetime in reliability test [34]. This is because it can easily approximate the normal distribution, logarithmic normal distribution and exponential distribution functions. In addition, it is also possible to analyze data even when two or more failure modes are present at the same time. The cumulative probability F(t) of a failure system can be introduced just as Weibull distribution function based upon a weakest-link model [34], which is expressed as ... 

---