Editing Sufficient statistic (section)

===Bernoulli distribution===

If ''X''<sub>1</sub>,&nbsp;....,&nbsp;''X''<sub>''n''</sub> are independent [[Bernoulli trial|Bernoulli-distributed]] random variables with expected value ''p'', then the sum ''T''(''X'') =&nbsp;''X''<sub>1</sub>&nbsp;+&nbsp;...&nbsp;+&nbsp;''X''<sub>''n''</sub> is a sufficient statistic for ''p'' (here 'success' corresponds to ''X''<sub>''i''</sub>&nbsp;=&nbsp;1 and 'failure' to  ''X''<sub>''i''</sub>&nbsp;=&nbsp;0; so ''T'' is the total number of successes)

This is seen by considering the joint probability distribution:

:<math> \Pr\{X=x\}=\Pr\{X_1=x_1,X_2=x_2,\ldots,X_n=x_n\}.</math>

Because the observations are independent, this can be written as

:<math>
p^{x_1}(1-p)^{1-x_1} p^{x_2}(1-p)^{1-x_2}\cdots p^{x_n}(1-p)^{1-x_n} </math>

and, collecting powers of ''p'' and 1&nbsp;−&nbsp;''p'',  gives

:<math>
p^{\sum x_i}(1-p)^{n-\sum x_i}=p^{T(x)}(1-p)^{n-T(x)}
</math>

which satisfies the factorization criterion, with ''h''(''x'')&nbsp;=&nbsp;1 being just a constant.

Note the crucial feature: the unknown parameter ''p'' interacts with the data ''x'' only via the statistic ''T''(''x'') =&nbsp;Σ&nbsp;''x''<sub>''i''</sub>.

As a concrete application, this gives a procedure for distinguishing a [[Fair coin#Fair results from a biased coin|fair coin from a biased coin]].