Given a model of some atomic positions, these positions and their respective Debye-Waller factors ( or B-factors, accounting for the thermal motion of the atom ) can be refined to fit the observed diffraction data, ideally yielding a better set of phases.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that assigns new examples into one category or the other.

Given the go-ahead, Marston developed Wonder Woman, basing her character on both Elizabeth and Olive Byrne, to be the model of an conventional, liberated, powerful modern woman.

Given local and transient influences on surface height, the values defined below are based on a " general purpose " model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level ( neglecting geoid height ).

Given the ACE model, researchers can determine what proportion of variance in a trait is heritable, versus the proportions which are due to shared environment or unshared environment.

Given the limited spectrum a wholesale model as noted by the EU Commission reference 2 in such situation is chosen.

Given the shortcomings of the linear regression model for dealing with a dichotomous criterion, it is necessary to use some other analysis.

Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit.

Given a set of training examples, each marked as belonging to one of two categories, an SVM training algorithm builds a model that predicts whether a new example falls into one category or the other.

Given a model of a system, exhaustively and automatically check whether this model meets a given specification.

Given a model M of a Skolem theory T, the smallest substructure containing a certain set A is called the Skolem hull of A.

Given a probability space, consider a single-period binomial model.

Given a set of data and the desire to produce some kind of “ model ” of those data ( model, in this case, meaning a function fitted through the data ), there are a variety of functions that can be chosen for the fit.

Given such a model, the sentence

However, the modelling study also states that " Given the complex, three-dimensional nature of the real Pine Island glacier ... it should be clear that the [...] model is a very crude representation of reality.

:" Given that the structure of an econometric model consists of optimal decision rules of economic agents, and that optimal decision rules vary systematically with changes in the structure of series relevant to the decision maker, it follows that any change in policy will systematically alter the structure of econometric models.

Given the assumptions of the model ( for example, that the profits are reinvested and that capital accumulation does not substitute for skilled labor in production ), the process becomes self-sustaining and leads to modernization and economic development.

Given a set of candidate models for the data, the preferred model is the one with the minimum AIC value.

Given a time series of data X < sub > t </ sub >, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series.

Given a good model, it is best to make as many observations as practicable, depending of the expected reliability of prior knowledge, cost of observations, time and resources available, and accuracy required.

Given a model selection problem in which we have to choose between two models, on the basis of observed data D, the plausibility of the two different models M < sub > 1 </ sub > and M < sub > 2 </ sub >, parametrised by model parameter vectors and is assessed by the Bayes factor K given by

Given any input, GEN generates an infinite number of candidates, or possible realizations of that input.

Given any random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >..., X < sub > n </ sub >, the order statistics X < sub >( 1 )</ sub >, X < sub >( 2 )</ sub >, ..., X < sub >( n )</ sub > are also random variables, defined by sorting the values ( realizations ) of X < sub > 1 </ sub >, ..., X < sub > n </ sub > in increasing order.

Given the antiquity of these sculptures and a technical refinement indicative of artists fully conversant with all the aspects of Greek sculpture, it has been suggested that Greek communities were directly involved in these realizations, and that " the area might be the cradle of incipient Buddhist sculpture in Indo-Greek style ".

Given samples from a population, the equation for the sample skewness above is a biased estimator of the population skewness.

Given the test scores of two random samples of men and women, does one group differ from the other?

Given that the ruins of the gas chambers at Birkenau have been washed by a column of water at least 35m in height based on climatological records since 1945, Markiewicz and his team were not optimistic at being able to detect cyanides so many years later ; nevertheless, having the legal permission to obtain samples, they collected some from areas as sheltered from the elements as possible.

Most methods of lipid extraction and isolation from biological samples exploit the high solubility of hydrocarbon chains in organic solvents. Given the diversity in lipid classes, it is not possible to accommodate all classes with a common extraction method.

Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

Given enough time, a hypothetical monkey typing at random would, as part of its output, almost surely produce all of William Shakespeare | Shakespeare's plays.

* Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.

* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

The quantum circuits used for this algorithm are custom designed for each choice of N and the random a used in f ( x ) = a < sup > x </ sup > mod N. Given N, find Q = 2 < sup > q </ sup > such that < math > N ^ 2

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

Given a probability space with, the indicator random variable is defined by if otherwise

Given N independent and identically distributed Rayleigh random variables with parameter, the maximum likelihood estimate of is

Given a random variate U drawn from the uniform distribution in the interval < nowiki >( 0, 1 )</ nowiki >, then the variate

Given a random variate U drawn from the uniform distribution in the interval < nowiki ></ nowiki >, the variate

: Given a random galaxy in a location, the correlation function describes the probability that another galaxy will be found within a given distance.

Given a random graph of n nodes and an average degree < math >< k ></ math >.

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance.

Fallacious evaluation: " Given that the counterattacks against Germany occurred only after they had conquered the greatest amount of territory under their control, regression to the mean can explain the retreat of German forces from occupied territories as a purely random fluctuation that would have happened without any intervention on the part of the USSR or the Western Allies.

Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value.

Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y.

Given a random sample ( X < sub > i </ sub >, Y < sub > i </ sub >), i = 1, ..., n, each pair ( X < sub > i </ sub >, Y < sub > i </ sub >) satisfies

Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

Given discrete random variable with support and with support, the conditional entropy of given is defined as:

Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, the distribution of G is approximately a chi-squared distribution, with the same number of degrees of freedom as in the corresponding chi-squared test.