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For a random sample as above, with cumulative distribution (), the order statistics for that sample have cumulative distributions as follows [2] (where r specifies which order statistic): () = = [()] [()] The proof of this formula is pure combinatorics: for the th order statistic to be , the number of samples that are > has to be between and .
K-distribution arises as the consequence of a statistical or probabilistic model used in synthetic-aperture radar (SAR) imagery. The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging. It is also ...
Cohen's kappa measures the agreement between two raters who each classify N items into C mutually exclusive categories. The definition of is =, where p o is the relative observed agreement among raters, and p e is the hypothetical probability of chance agreement, using the observed data to calculate the probabilities of each observer randomly selecting each category.
As with the ¯ and s and individuals control charts, the ¯ chart is only valid if the within-sample variability is constant. [4] Thus, the R chart is examined before the x ¯ {\displaystyle {\bar {x}}} chart; if the R chart indicates the sample variability is in statistical control, then the x ¯ {\displaystyle {\bar {x}}} chart is examined to ...
The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V.
for k = 0, 1, 2, ..., n, where =!! ()! is the binomial coefficient. The formula can be understood as follows: p k q n−k is the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are "successes" and the remaining n − k trials
The multinomial distribution models the outcome of n experiments, where the outcome of each trial has a categorical distribution, such as rolling a k-sided die n times. Let k be a fixed finite number. Mathematically, we have k possible mutually exclusive outcomes, with corresponding probabilities p 1, ..., p k, and n independent trials.
The χ 2 distribution given by Wilks' theorem converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of choosing the fixed log-likelihood difference is to make the confidence acceptably high while keeping the region acceptably small (narrow range of estimates).