Distribution hypergeometric

Nature In an experiment in which one samples from a relatively small group of items, each of which is classified in one of two categories, A or B, the hypergeometric distribution can be defined. One example is the probabihty of drawing two red and two black cards from a deck of cards. The hypergeometric distribution is the analog of the binomial distribution when successive trials are not independent, i.e., when the total group of items is not infinite. This happens when the drawn items are not replaced. 

If the probabilities do not remain constant over the trials and if there are k (rather than two) possible outcomes of each trial, the hypergeometric distribution applies. For a sample of size N of a population of size T, where... 

Therefore, exact tests are considered that can be performed using two different approaches conditional and unconditional. In the first case, the total number of tumors r is regarded as fixed. As a result the null distribution of the test statistic is independent of the common probability p. The exact conditional null distribution is a multivariate hypergeometric distribution. 

Fig. 7 Transcriptome effects of T3 administration on the developing zebrafish embryo, (a) Heatmap of microarray results from 652 probes showing significant differences between T3-treated and control samples. Fold induction values (in log scale) are represented by different shades of color (scale shown in the far-right bar.), (b) Distribution of overrepresented (red), underrepresented (blue), and unchanged/undetected (ivory) transcripts in T3-treated embryos belonging to the functional categories ossification, visual processes, and oxygen transport. The significance of the observed variations (p values) was calculated by the hypergeometric distribution with the Bonferroni correction...

Note the relation to the hypergeometric distribution (1.1.6). This p coincides with pi provided that a and ft are connected by the detailed balance relation (4.4), where hv is the energy gap, energy differences inside each band being neglected. 

The hypergeometric distribution can be generalized to a multivariable form, the multivariate hypergeometric distribution, which can be used to extend Fisher s Exact Test to contingency tables larger than 2 by 2 and to multidimensional contingency tables. There is statistical software available to perform these calculations however, due to the complexity of the calculations and the large number of trial tables whose probability of occurrence must be calculated, this extension has received limited use. 

A number of other discrete distributions are listed in Table- 1.1, along with the model on which each is based. Apart from the mentioned discrete distribution of random variable hypergeometrical is also used. The hypergeometric distribution is equivalent to the binomial distribution in sampling from infinite populations. For finite populations, the binomial distribution presumes replacement of an item before another is drawn whereas the hypergeometric distribution presumes no replacement. 

Large populations of data (in excess of 5000 items) can be regarded as infinite and thus a binomial approximation to the hypergeometric distribution can be applied. It is assumed that errors occur randomly throughout the data population. If data within the population has been obtained from different sources in different ways, there may be an expectation that error rates for these subpopulations may differ. If this is the case, the data population should be split into strata and analyzed separately. Note that for populations less than 5000 items it is recommended that all items be checked rather than a sample taken. 

If N is much less than T then the hypergeometric distribution becomes indistinguishable from the binomial distribution where =P(1-P)N and P is simply F/T. This condition certainly holds when the entire file is used as a reference. It is even a good approximation when only the training set is used since the actives are much less numerous than the inactives. 

Table 10.7, the probability of the distribution of responses in the 2x2 table is calculated from the hypergeometric distribution as ... 

To obtain values for the expected and maximum values, the hypergeometric distribution is assumed. If Uij is the number of genes that are common to clusters i and j from each method, the adjusted Rand index can be calculated from the following formula ... 

The upper bound of the adjusted Rand index is unity and it takes on the value of zero when the similarity between the two clustering methods matches the expectation of the hypergeometric distribution. A higher value of the adjusted Rand index indicates a greater similarity of membership between the clusters of the two methods being compared. 

Fig. 6. Enrichment analysis results for clozapine targets against GeneGo Pathway Maps and GeneGo Biological Processes. Horizontal bars indicate the negative log of the p-value from the hypergeometric distribution calculation.

Bates, J. W., cuid J. A. Lcunbert. 1991. Use of the hypergeometric distribution for sampling in forensic glass comparison. Journal of the Forensic Science Society 31 (4) 449-455. 

The UNODC guidelines for sampling are based upon the statistical method known as hypergeometric distribution. Hypergeometric distribution is used to determine the probability of the number of positive samples likely to be found in a certain population. This is based on the following equation ... 

If the population is limited in size, sampling without replacement leads to exhaustion of the population. Then the hypergeometric distribution is applied to calculate the chances P(x). However differences are mostly small or negligible and will not be further be discussed here. 

Microsoft Excel has a built-in hypergeometric distribution function, HYPERGEOMDIST, that calculates probability. 

Gaussian curve/distribution Grubbs test Homogeneous Hypergeometric distribution Hypothesis... 

Use the Excel function for the hypergeometric distribution to calculate the probability of selecting five seimples that wfll be positive for cocaine if a seizure contains 1000 baggies and past laboratory data suggest that 50% of such recent seizures are cocaine and the rest are not. Study the results and explain how the hypergeometric distribution is similar to a simple binomial distribution. 

Weisstein, E. W. "Hypergeometric Distribution." Mathworld, June 12,2004. On the Internet at http //mathworld. wolfram.com. 

Hypergeometric distribution A distribution of possible outcomes of an event or events when samples are withdrawn or replaced. 

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