grandes-ecoles 2020 Q21

grandes-ecoles · France · centrale-maths2__pc Probability Generating Functions Uniqueness and characterization of distributions via PGF

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.

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.

Paper Questions

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