Definition:Sufficient Statistic

Definition
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population indexed by a parameter $\theta$.

Let $T$ be a statistic.

Let $I = \Img {\map T {X_1, X_2, \ldots, X_n} }$.

Let $D$ be the conditional joint distribution of $X_1, X_2, \ldots, X_n$ given $T = t$ and $\theta$.

We call $T$ a sufficient statistic for $\theta$ if $D$ is independent of the value of $\theta$ for all $t \in I$.