Normal distribution weighting factor

F(x), here, is (Es + AEs)/ Er + A )), as we just noted. In the previous case, the weighting function was the Normal distribution. Our current interest is the Poisson distribution, and this is the distribution we need to use for the weighting factor. The interest in our current development is to find out what happens when the noise is Poisson-distributed, rather than Normally distributed, since that is the distribution that applies to data whose noise is shot-noise-limited. Using P to represent the Poisson distribution, equation 49-59 now becomes... 

Figure 9- Log-normal distribution of HDPE molecular weight after 1.5 h at 165 °C and 10 h at 135 °C 1 (upper line) - HDPE with antioxidant circles - HDPE without antioxidant 2 (lower line) upper broken line has been deplaned vertically by a factor of 2.2 to coincide with the low molecular weight data of unstabilized sample.

In other words, the experimental determinations made towards the end of a reaction contribute very little to the least squares estimate of the rate coefficient. Eqn. (68) is of little use in computation, however, unless we can make some statement about ff(ay). There are two clearcut situations which we can discuss. The ffist of these situations is that the value of ff(ay) does not depend on J and so within any one run, ff(ay) is a constant. In other words, were it possible to determine a large number of values of each ay, the width of the normal distribution would be the same in each case. Since it is only relative weights which are important in computation, we can omit such constant factors from the weights [as was done in eqn. (65)] and write... 

The weighting factor m( y) is very similar to a gaussian function centered on y, as illustrated in Fig. 4.3. Thus the van Bemmelen-Freundlich isotherm can be thought of as the result of a log-normal distribution of Langmuir parameters iC (i.e, a normal distribution of In K) in a soil. The parameter (3 determines the narrowness of this distribution, in that ° ... 

For the fitness function a normalization of the difference between simulated and measured data is used in to get a uniform distribution of the deviations at low and high impedance values. With the weighting factor H N - 1) the results get more independence on the exact number of frequencies. 

The best-fit estimates of the charge transfer parameters are calculated as the weighted mean of the values obtained at the four rotation speeds taken into consideration. Introducing weighting factors helps to adjust the impact of one result on the mean with regard to their level of uncertainty. The maximum likelihood estimate of normally distributed variables x with different variances of and the same mean value i is given by (Sorenson, 1980) ... 

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