![]() ![]() The lower row shows how the presence of clustering results in the data points deviating from the corresponding Poisson distributions. The upper row of Figure 1 illustrates near ideal random distributions of defects observed across the microchips of two individual silicon wafers, for moderate and high densities of defects, respectively. genetic, nutrition, educational factors). ![]() It should be noted that the factors causing such additional within-clusters variation are not observable and correspond to characteristics of the population in the clusters (e.g. Clustering is itself a random effect that occurs within a restricted area or interval. The clustering expresses the characteristic by which the occurrence of events – in this case defects – within a unit area, increases beyond that which would be expected from a random process alone. In each panel, the green curve shows the negative binomial fit to the data, while in the background, the grey shaded area depicts the limit of the Poisson distribution for the corresponding average defect density. The cumulative frequency plots in Figure 1 demonstrate how the negative binomial can adapt to random or clustered data of various densities. It can be considered as a generalization of the Poisson distribution, without the requirement of a constant mean density that also equals the variance. The negative binomial distribution is especially useful when describing distributions where the underlying density varies continuously, according to a gamma distribution. More generally, the negative binomial is often the leading choice for modelling discrete frequency distributions – whether that is counting insects on a sample of leaves, accidents experienced by workers, or yeast cell counts across a haemocytometer. In the semiconductor industry, the negative binomial distribution has long been a favorite of engineers who need to model how a change in the rate of manufacturing defects will affect the number of functioning microchips. What is the negative binomial distribution? ![]()
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