Editing Sufficient statistic (section)

===Poisson distribution===

If ''X''<sub>1</sub>,&nbsp;....,&nbsp;''X''<sub>''n''</sub> are independent and have a [[Poisson distribution]] with parameter ''λ'', then the sum ''T''(''X'') =&nbsp;''X''<sub>1</sub>&nbsp;+&nbsp;...&nbsp;+&nbsp;''X''<sub>''n''</sub> is a sufficient statistic for&nbsp;''λ''.

To see this, consider the joint probability distribution:

:<math>
\Pr(X=x)=P(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>
{e^{-\lambda} \lambda^{x_1} \over x_1 !} \cdot
{e^{-\lambda} \lambda^{x_2} \over x_2 !} \cdots
{e^{-\lambda} \lambda^{x_n} \over x_n !}
</math>

which may be written as

:<math>
e^{-n\lambda} \lambda^{(x_1+x_2+\cdots+x_n)} \cdot
{1 \over x_1 ! x_2 !\cdots x_n ! }
</math>

which shows that the factorization criterion is satisfied, where ''h''(''x'') is the reciprocal of the product of the factorials.  Note the parameter λ interacts with the data only through its sum ''T''(''X'').