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# Statistical Treatment of Parallel Elements

IP.com Disclosure Number: IPCOM000085041D
Original Publication Date: 1976-Feb-01
Included in the Prior Art Database: 2005-Mar-02
Document File: 3 page(s) / 64K

IBM

## Related People

Robbins, GJ: AUTHOR [+2]

## Abstract

The statistical behavior of an assembly of elements can be simplified by considering groups of series elements and groups of parallel elements. The treatment of series elements is well-known (e.g., the standard deviation of the total delay in a path of independent elements is the square root of the sum of the squares of each element's standard deviation). The behavior of parallel elements, however, requires special treatment since serial compensation of properties (e.g., "fast" elements compensating for "slow" elements) does not occur. A method of treating the statistical behavior of parallel elements is described herein.

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Statistical Treatment of Parallel Elements

The statistical behavior of an assembly of elements can be simplified by considering groups of series elements and groups of parallel elements. The treatment of series elements is well-known (e.g., the standard deviation of the total delay in a path of independent elements is the square root of the sum of the squares of each element's standard deviation). The behavior of parallel elements, however, requires special treatment since serial compensation of properties (e.g., "fast" elements compensating for "slow" elements) does not occur. A method of treating the statistical behavior of parallel elements is described herein.

Consider an element which has a probability distribution function (PDF), p(x) for a property x. Such a PDF is characterized by a mean (u(x)) and a standard deviation (Sigma(x)). The element statistic p(x) can be used to generate a cumulative distribution function (CDF), P(x), which describes the probability of the property x exceeding any value X (by integrating p(x) from that value to + Infinity) or that x is less than any value X (by integrating p(x) from - Infin.) to that value). Figs. 1 and 2 show a PDF and CDF for a hypothetical element.

If a combination of Eta such elements is connected in parallel, then the probability that the assembled property x(o) equals or exceeds X is given by the cumulative binomial distribution:

(Image Omitted)

For all values of X, this resultant CDF for the parallel assembly may be differentiated to yield the PDF of the property x(o) for the parallel assembly.

The case where all elements are statistically equivalent but independent is sometimes of interest. The following TABLE 1 shows the results of applying this treatment to Eta elements in parallel, where each element is statistically equivalent and independent with both a normal distribution and a 3-sigma truncated normal distribution - both described by a mean Mu and standard deviation Sigma.

The parameter Alpha is defined to relate the shift in the mean of the resulting distributio...