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

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 the model and realizations ( samples ) of the random vector, the task is to estimate both the mixing matrix and the sources.

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.

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

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 sample, the expectation function is approximated by the sample average

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.