less tersely, suppose formula_133 are independent identically distributed random variables whose distribution is known to be in some family of probability distributions.
Anglais
less tersely, suppose formula_146 are independent identically distributed random variables whose distribution is known to be in some family of probability distributions.
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Espagnol
in particular, in euclidean space, these conditions always hold if the random variables (associated with formula_63 ) are all discrete or are all continuous.
Anglais
in particular, in euclidean space, these conditions always hold if the random variables (associated with formula_76 ) are all discrete or are all continuous.
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Espagnol
==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.
Anglais
==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.
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