Given a fixed budget, this can allow an increased sample size.

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 the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

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

Given a sample of wood, the variation of the tree ring growths provides not only a match by year, it can also match location because the climate across a continent is not consistent.

Given a sample of DNA, the instrument determines the order of the four bases — adenine, guanine, cytosine, and thymine — composing pieces of the sample.

Given the observed percentage difference p − q ( 2 % or 0. 02 ) and the standard error of the difference calculated above (. 03 ), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0. 02 and standard deviation 0. 03 is greater than 0.

Given any set of points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian.

Given a sufficiently large sample size, a statistical comparison will always show a significant difference unless the population effect size is exactly zero.

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

Given a sample consisting of n independent observations x < sub > 1 </ sub >,..., x < sub > n </ sub > of a p-dimensional random vector X ∈ R < sup > p × 1 </ sup > ( a p × 1 column-vector ), an unbiased estimator of the ( p × p ) covariance matrix

Given a sample of n independent observations x < sub > 1 </ sub >,..., x < sub > n </ sub > of a p-dimensional zero-mean Gaussian random variable X with covariance R, the maximum likelihood estimator of R is given by

Given the estimation of from, the error bars of can be estimated by the sample variance using the unbiased estimate of the variance:

Given a random sample of T observations from this process, the ordinary least squares estimator is

Given the sample mean () and sample coefficient of variation () the parameters and can be estimated:

: Given that there are almost eight times as many data points in this sample as in the previous analysis by Burbidge & Napier ( 2001 ), we must conclude that the previous detection of a periodic signal arose from the combination of noise and the effects of the window function.

Given that v < sup > 2 </ sup > ignores direction, it is logical to assume that the formula can be extended to the entire sample, replacing m with the entire sample's mass, equal to the molar mass times the number of moles n yielding

Given the sample ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, …, x < sub > n </ sub >), it is natural to estimate the characteristic function as

Given a total sample size greater than 1.

Given the wide expectation that the war was going to be a short one, the majority however supported the war effort and the call to restore the " freedom of small nations " on the European continent.

" Given the ubiquity of keyboards and the growing expectation that secondary school pupils and university students will type their essays and coursework, I think it ( touch typing ) is one the most useful skills a child can learn-at any age from seven upwards.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given a function f of type, currying it makes a function.

Given a function of type, currying produces.

Given the definition of above, we might fix ( or ' bind ') the first argument, producing a function of type.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given a function f of a real variable x and an interval of the real line, the definite integral

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

Given a relation, scaling the argument by a constant factor causes only a proportionate scaling of the function itself.

# Composition operator ( also called the substitution operator ): Given an m-ary function and m k-ary functions:

# Primitive recursion operator: Given the k-ary function and k + 2-ary function:

# Minimisation operator: Given a ( k + 1 )- ary total function:

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that