## Statistics   ## Conjecture Sur Les Familles Exponentielles

in this article we will establish some properties of random variables and then we will propose a conjecture related to the exponential family. This conjecture seems interesting to me. Our results are based on the consideration of continuous random variables $X_{i}$ defined on the same space $\Omega$ and the same super-extra density law of parameter $\theta_{i}$ and canonique function $T$ Let $n\in \mathbb{N}^{*}$ Considering the random variable $J$ and $I$ a subsect of $\{1,2,..n\}$ such that : $X_{J}=\inf_{i\in I}(X_{i})$ we show that : $$\forall i\in I:\mathbb{P}( J=i)=\frac{\theta_{i}\prod_{j\in I}c(\theta_{j})}{\sum_{j\in I}\theta_{j}}\int_{T(\Omega)}e^{-x}dx$$. We conjecture that if the density of $X_{i}$ is $c(\theta_{i})e^{-\theta_{i}T(x)}\mathbf{1}_{\Omega}(x)$ Hence $\exists h,r$ two functions h such that $$\forall i\in I:\mathbb{P}( J=i)=\frac{r(\theta_{i})\prod_{j\in I}h(\theta_{j})}{\sum_{j\in I}r(\theta_{j})}\int_{T(\Omega)}e^{-x}dx$$