We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$. Using the results from the preamble, show that, for all $t \in ] - R _ { X } , R _ { X } [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation and that $M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right)$.
We assume that, for all non-zero natural integer $n$, $X$ admits a moment of order $n$ and that the power series $\sum _ { n \geqslant 0 } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$ has a radius of convergence $R _ { X } > 0$. For all $t \in ] - R _ { X } , R _ { X } [$, we denote $M _ { X } ( t ) = \sum _ { n = 0 } ^ { + \infty } m _ { n } ( X ) \frac { t ^ { n } } { n ! }$.
Using the results from the preamble, show that, for all $t \in ] - R _ { X } , R _ { X } [$, the random variable $\mathrm { e } ^ { t X }$ admits an expectation and that $M _ { X } ( t ) = \mathbb { E } \left( \mathrm { e } ^ { t X } \right)$.