Results for generating correlated random vari... translation from Spanish to English

less tersely, suppose formula_133 are independent identically distributed random variables whose distribution is known to be in some family of probability distributions.
less tersely, suppose formula_146 are independent identically distributed random variables whose distribution is known to be in some family of probability distributions.

in particular, in euclidean space, these conditions always hold if the random variables (associated with formula_63 ) are all discrete or are all continuous.
in particular, in euclidean space, these conditions always hold if the random variables (associated with formula_76 ) are all discrete or are all continuous.

==examples=====bernoulli distribution===if "x"1, ..., "x""n" are independent bernoulli-distributed random variables with expected value "p", then the sum "t"("x") = "x"1 + ... + "x""n" is a sufficient statistic for "p" (here 'success' corresponds to "x""i" = 1 and 'failure' to "x""i" = 0; so "t" is the total number of successes)this is seen by considering the joint probability distribution::formula_66because the observations are independent, this can be written as:formula_67and, collecting powers of "p" and 1 − "p", gives:formula_68which satisfies the factorization criterion, with "h"("x") = 1 being just a constant.
==examples=====bernoulli distribution===if "x"1, ..., "x""n" are independent bernoulli-distributed random variables with expected value "p", then the sum "t"("x") = "x"1 + ... + "x""n" is a sufficient statistic for "p" (here 'success' corresponds to "x""i" = 1 and 'failure' to "x""i" = 0; so "t" is the total number of successes)this is seen by considering the joint probability distribution::formula_79because the observations are independent, this can be written as:formula_80and, collecting powers of "p" and 1 − "p", gives:formula_81which satisfies the factorization criterion, with "h"("x") = 1 being just a constant.