We assume in this question that $X ( \Omega )$ is a finite set of cardinality $r \in \mathbb { N } ^ { * }$.\\
We denote $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ where the $x _ { i }$ are pairwise distinct, and, for all integer $k \in \llbracket 1 , r \rrbracket , a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all real $t , \phi _ { X } ( t ) = \sum _ { k = 1 } ^ { r } a _ { k } \mathrm { e } ^ { \mathrm { i } t x _ { k } }$.