Differential frequency distributions

The different MW averages are derived by using the differential or frequency distribution curves and are reported as number average MW (Mn), weight average... 

More often in practical measurements x and its distribution function are continuous variables and therefore F(x) may be differentiated to give a (probability) density function f(x) the shape of which resembles the frequency distribution. Further details are not of interest here, but we should know that we utilize such density functions via well-known statistical tables. 

During the mixing process, for a at any time a frequency distribution (differential or density distribution) /(o) ensues. After the area normalization, the integral of the frequency distribution is equal to 1 ... 

Then, distributions for the occurrences of the various molecular properties in two sets of molecules, one of active molecules and the other of inactive molecules, are evaluated. From these frequency distributions, weights are calculated using one of the two weighting schemes, each of which seeks to quantify the differential occurrences of the defined ranges of property values in active (AC) and inactive (IN) molecules ... 

The frequency distribution is thus a differential quantity, and it is given in number per unit volume per unit of x. If x is expressed in units of length, e.g., particle diameter, the dimension of/(x) is [L-4]. 

Obviously, the details in the time-profile, 7, and the frequency spectrum, Fp, of the incident X-pulse, depend on the experimental setup. However, if the duration of the pulse is either sufficiently short or sufficiently long compared to the time scale of the nuclear dynamics, 7 may be replaced by either a delta function or a constant on the nuclear time scale. Likewise, if the width of Fp can be neglected (known as the static approximation ), we can obtain simplified expressions for the differential scattering signal. However, as pointed out earlier, the frequency widths of X-ray pulses obtained from, e.g., synchrotron radiation are typically on the order of percent of the carrier frequency. Hence, in order to simulate the finer details of the experimental signal, the actual frequency distribution of the incident X-ray pulse must be taken into account [29],... 

Distribution functions can be classified as discontinuous or continuous. Discontinuous distribution functions are subdivided into frequency distributions and cumulative distributions. Continuous distribution functions are further classified as differential and integral distribution functions. 

Figure 8-1. Frequency distribution (top left) and a cumulative distribution (bottom left), together with a differential distribution (top right) and an integral distribution (bottom right) of the mole fraction x of a property E (diagrammatic). Read Jjcd instead of Sxdx.

Discontinuous distribution functions can, of course, be transformed into continuous distribution functions when the difference between two neighboring properties is very small compared with the whole range of property values. Frequency distributions convert to the corresponding differential distributions and cumulative distributions convert to integral distributions,... 

When such a particle passes through a thin absorber (foil) with plasma frequency cOpi and enters a medium with cOp2 it radiates photons of frequency co cOpi with a double differential energy distribution... 

Figure 1.3 Typical differential and cumulative frequency distributions...

The first step is to estimate the number of particles per unit mass of A and B. This is done by converting the size distributions into differential frequency number distributions and using ... 

The resulting frequency distribution h(i) of chain lengths i can be derived by solving the system of differential equations sequentially starting with [Pi]t=o = [f]o and the definition of the kinetic chain length given above [Eq. (24)]. 

Figure 6. Frequency distribution of traffic accidentswith the presence of a hazardous substance on the road for the years 2007-2010, differentiated according to regions.

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