CDF

Fig. 2. Empirical cdf s for many samples of increasing size from a uniform distribution...

For the robust estimation of the pair potentials, some obstacles had to be overcome. There are a huge number of different triples (si, Sk,i — k), and to find densities, we needed a way to group them in a natural way together into suitable classes. A look at the cumulative distribution functions (cdf s) of the half squared distances Cjfc at residue distance d = i — k (w.l.o.g. >0), displayed in Figure 1, shows that the residue distances 8 and higher behave very similarly so in a first step we truncated all residue distances larger than 8 to 8. 

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

An independent development of a high pressure polymerization technology has led to the use of molten polymer as a medium for catalytic ethylene polymerization. Some reactors previously used for free-radical ethylene polymerization at a high pressure (see Olefin polymers, low density polyethylene) have been converted to accommodate catalytic polymerization, both stirred-tank and tubular autoclaves operating at 30—200 MPa (4,500—30,000 psig) and 170—350°C (57,83,84). CdF Chimie uses a three-zone high pressure autoclave at zone temperatures of 215, 250, and 260°C (85). Residence times in all these reactors are short, typically less than one minute. 

For this duty cycle, the abbreviation is followed by the indication of the cyclic duration factor, the number of duty cycles per hour (c/h) and the factor of inertia (Ff). (See Section 3.4 for FI.) Thus, for a 40% CDF with 90 operating cycles per hour and factor of inertia of 2.5, the cycle will be represented by... 

We have considered three different speeds (lower Ni to higher /V/ ,) for this duty cycle, having three CDF , for one cycle, each corresponding to a different speed. 

As an illustration the CDF must be indicated for each speed as follows ... 

Note The above example is only for a general reference. The CDF of the transformer, for the short-time rating, should be increased writh the starting time and the number of starts per hour. Refer to the transformer manufacturer for a more appropriate selection. 

The CDF can be controlled by controlling the period of conduction, in other words, the pulse widths (periodic time period, T remaining the same). Thus the a.c. output voltage in an IGBT inverter can be controlled with the help of modulation. The modulation in the inverter circuit is acliieved by superposing a cairier voltage waveform... 

Figure 4.6 Shape of the Cumulative Distribution Function (CDF) for an arbitrary normal distribution with varying standard deviation (adapted from Carter, 1986)...

When the loading stress ean be determined by a CDF in elosed form, this simplifies to ... 

This ean also be simplified when the strength CDF is in elosed form to give ... 

Bury (1974 1975 1978 1999) introduces the concept of duty cycles to static design in a different approach. The duty cycle or mission length of a design is equivalent to the number of load applications, however, it is only the maximum value which is of design significance. If the load, L, is a random variable then so is its maximum value, L, and the PDFs of each are related. It can be shown that the CDF, F L), of the maximum among independent observations on L is ... 

For example, to determine the reliability, R , for independent load applications, we can use equation 4.33 when the loading stress is modelled using the Maximum Extreme Value Type I distribution, as for the above approach. The CDF for the... 

Therefore, the loading stress CDF ean be represented by a 3-parameter Weibull distribution ... 

R = reliability at th applieation of load n = number of independent load applieations in sequenee F L) = loading stress CDF f S) = strength PDF. 

In the problem here, the loading stress is a Normal distribution and the strength is a 3-parameter Weibull distribution. Beeause the Normal distribution s CDF is not in elosed form, the 3-parameter Weibull distribution ean be used as an approximating distribution when [3 = 3.44. The parameters for the 3-parameter Weibull distribution. 

Typically, the random values, x, from a particular distribution are generated by inverting the closed form CDF for the distribution F x) representing the random variable, where ... 

Many distributions can be represented in closed form except for the Normal and Lognormal types. The CDF for these distributions can only be determined numerically. For example, the 3-parameter Weibull distribution s CDF is in closed form, where ... 

Therefore, for the 3-parameter Weibull distribution, the inverse CDF with respect to u is ... 

In Seetion 4.4.1 we deseribed a means of determining the reliability, R, when the loading stress, L, is represented by CDF in elosed form (for all the distributions exeept for the Normal and Lognormal type) and the strength, S, is represented by its PDF. 

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