In this paper we explain a necessary and sufficent condition for independence between any arbitrary statistics with sufficient statistics which is also maximum likelihood estimator in a general‎
‎exponential family with location and scale parameter namely generalized normal distribution‎. ‎At the end‎, ‎it is shown that the converse is true except in the asymptotic cases‎.

‎In this paper‎, ‎after introducing exponential family and a history of work done by researchers in the field of statistics‎, ‎some applications of this family in statistical inference especially in estimation problem‎,‎statistical hypothesis testing and statistical information theory concepts will be discussed‎.