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... 

After substitution of (A3.1) into (6.14), several integrals of the same type must be calculated. These integrals can be expressed via the degenerate hypergeometrical function d>(-, -, -) and gamma-function T( ) ... 

Although the hypergeometric functions are useful in spectroscopy, as they describe the rotation of a symmetric top molecule (Section 9.2.4), their importance is primarily due to their generality. If, for example, a = 1 and fi say, Eq. (154) becomes a +i — a for all values of n. The result is the ordinary geometric series... 

The Chebyshev polynomials, whiGh occur in quantum chemistry and in certain numerical applications, can be obtained from the hypergeometric functions by placing a = -/ , an integer, and y — Finally, the hypergeometric... 

The resulting series is a particular solution to Eq. (153) known as the hypergeometric series. It converges for x] < 1. It is usually denoted as... 

A sequence of approximations, using properties of the confluent hypergeometric function, integration by steepest descents, and judicious discard of all but the dominant terms, gives one the asymptotic form... 

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...

A hypergeometric probability (HP) can be used to quantitatively measure the accuracy of a soil geochemistry program tested over a mineral deposit target with... 

By direct integration or by specializing overlap coefficients between alternative harmonics [25] we are able to write it directly as a single sum of the Racah type. This sum [26] is a hypergeometric function 4F3 of unit argument ... 

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