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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$. Let $Y : \Omega \rightarrow \mathbb { R }$ be a random variable satisfying $Y ( \Omega ) = X ( \Omega )$ and, for all $n \in \mathbb { N }$, $$\mathbb { P } \left( Y = x _ { n } \right) = \frac { a _ { n } x _ { n } ^ { 2 k } } { \alpha }$$ Show that $\phi _ { Y }$ is of class $C ^ { 2 }$ on $\mathbb { R }$.
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 $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$.\\
We fix a natural integer $k \in \mathbb { N }$ and assume both that $\phi _ { X }$ is of class $C ^ { 2 k + 2 }$ on $\mathbb { R }$ and that $X$ admits a moment of order $2 k$. We denote $\alpha = \mathbb { E } \left( X ^ { 2 k } \right)$ and assume that $\alpha > 0$.\\
Let $Y : \Omega \rightarrow \mathbb { R }$ be a random variable satisfying $Y ( \Omega ) = X ( \Omega )$ and, for all $n \in \mathbb { N }$,
$$\mathbb { P } \left( Y = x _ { n } \right) = \frac { a _ { n } x _ { n } ^ { 2 k } } { \alpha }$$
Show that $\phi _ { Y }$ is of class $C ^ { 2 }$ on $\mathbb { R }$.