Let $X$ and $X^{\prime}$ be two random variables taking values in $\mathbb{N}$. Justify that $X \sim X^{\prime}$ if and only if $G_{X} = G_{X^{\prime}}$.

Let $X$ and $X^{\prime}$ be two random variables taking values in $\mathbb{N}$. Justify that $X \sim X^{\prime}$ if and only if $G_{X} = G_{X^{\prime}}$.

Let $X : \Omega \rightarrow \mathbb { R }$ and $Y : \Omega \rightarrow \mathbb { R }$ be two discrete random variables such that $\phi _ { X } = \phi _ { Y }$. Show that, for all $m \in \mathbb { R } , \mathbb { P } ( X = m ) = \mathbb { P } ( Y = m )$, in other words that $X$ and $Y$ have the same distribution.