Summing random

The stochastic tools used here differ considerably from those used in other fields of application, e.g., the investigation of measurements of physical data. For example, in this article normal distributions do not appear. On the other hand random sums, invented in actuary theory, are important. In the first theoretical part we start with random demand and end with conditional random service which is the basic quantity that should be used to decide how much of a product one should produce in a given period of time. 

Three concepts are important random demand, random sums and conditional random demand. 

If, for example, a product has independent daily random demand given by the gamma density with parameters X, c then the weekly demand is described by the gamma density with parameters X, 7c or X, Sc depending on the behavior of demand at weekends. In many cases gamma densities are a good approximation of random demand. If, however, for a product the amounts vary significantly from order to order, it may be necessary to consider the demand as a random sum. 

Compound densities, also called random sums, are typically applied in modeling random demands or random claim sums in actuarial theory. The reason for this is simple. Assume that customers randomly order different quantities of a product. Then the total quantity ordered is the random sum of a random number of orders. The conversion to the actuarial variant is obvious The total claim is made from the individual claims of a random number of damage events. Zero quantities usually are neither ordered nor claimed. This leads to the following definition. 

Random Sum and Gamma Density with Equal Mean and Standard Deviation... 

T Random Sum Gamma Density I Fig. 6.2 Compound and gamma demand I (density). 

Note that the mean /i (8) has a proportional influence on the standard deviation of the total ordered amount. The tendency is intuitive The formula means that only a few large orders require more safety stock to cover random demand variations than many small orders. It is therefore a good idea to measure the mean and the variance of demands twice. First in the usual way as mean and variance of a sequence of, say weekly, figures and then by analyzing the orders of a historic period and applying Eq. (6.6). The comparison of the two results obtained often provides insight in the independence and the randomness of the historic demand. If the deviation of the two mean values and/or two variances is large then the demand can not be considered as a random sum. A reason could be, for example, that the demand... 

The final uncertainty of the fit was calculated as a random sum of differences of the fits without function h and with h Z) from the binomial expansion of the expression... 

Games of chance inspired several early investigations of the random sum representing the number of successes in n Bernoulli trials ... 

Consider a sequence of independent identically distributed time random variables, T. Ti. the probability that any single one of the Ts satisfies t < T < t + dt will be denoted by q(t)dt where q(t) is the pdf of the time between successive steps or the pausing time density. The time intervals T n) at which the rath step is taken are given by the random sum... 

Now let us suppose that in a time interval [0, t] the random walker executes n steps, and then the displacement of the walker is the random sum... 

In this section, the algebra of random variables will be rehearsed from the viewpoint of error propagation. Convolution, central limit theorem, as well as random sums (i.e., sums of a random number of random variables) are also included here because of their importance in nuclear applications. 

Sums of a random number of random variables - in short random sums - are often encountered at several stages of radiation detection. The problem usually presents itself disguised as a product. 

This asymmetrical result implies that if the (expected) number of terms is large enough, then the variance of the random sum is principally determined by the variance of the number of these terms (i.e., the variance of the individual terms is of relatively less importance). It is worthwhile to compare this result with Eq. (9.61), showing the formula of error propagation... 

As shown earlier, O Eq. (9.56) gives a different error formula for a different type of product. In the case of products therefore it is worthwhile to think it over whether or not a random sum is hiding behind the problem. This can only happen, of course, when one of the variables is an integer having no dimension. 

Note that the above equation shows a random sum. According to the results discussed at renewals (see O Eq. (9.117)), the random variable N is asymptotically normal with the... 

Relevant mechanisms that cause loss of counts from the photopeak include partial conversions of the incident energy event with part of the energy escaping the detector, true coincidence summing, random. summing of pulses, also known as pile-up, and dead time. While the first two effects are part of the detector s efficiency and are the same for all sample and standard counts, random summing and dead time are dependent on a sample s activity. ... 

Pile-up rejection is useful for reducing random summing at moderate to high count rates. At low count rates, it should be switched out. 

If counting losses due to random summing and/or self-absorption are corrected for, then these corrections will themselves have an uncertainty that must be accounted for. If these corrections are made by the spectrum analysis program, you should make sure, by reading the manual and by validation measurements, that the uncertainties assigned by the program are reasonable. 

Random summing 0.2 Standard Normal 1 0.2 Accounted post-analysis... 

In Chapter 4, I discussed random summing in connection with the pile-up rejection circuitry in amplifiers. We came to the conclusion that even with pile-up rejection there must be some residual random coincidences. There is then, whether or not pile-up rejection is available, a need to be able to correct for random summing in high count rate spectra. In some circles, there seems to be an assumption that pile-up rejection is 100% effective... 

If we rearrange this, we can derive a simple equation to correct peak areas for random summing ... 

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