## Conjecture Sur Les Familles Exponentielles

**Authors:** Idriss olivier BADO

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
$$ \forall i\in I:\mathbb{P}( J=i)=\frac{r(\theta_{i})\prod_{j\in I}h(\theta_{j})}{\sum_{j\in I}r(\theta_{j})}\int_{T(\Omega)}e^{-x}dx$$

**Comments:** 5 Pages.

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### Submission history

[v1] 2019-05-18 15:08:29

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