grandes-ecoles 2020 Q30

grandes-ecoles · France · centrale-maths2__pc Probability Generating Functions Deriving moments or distribution from a PGF

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Deduce an expression of $\mathbb { E } \left( X ^ { k } \right)$ in terms of $\phi _ { X } ^ { ( k ) } ( 0 )$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$.\\
Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$.\\
Deduce an expression of $\mathbb { E } \left( X ^ { k } \right)$ in terms of $\phi _ { X } ^ { ( k ) } ( 0 )$.

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