grandes-ecoles 2020 Q6

grandes-ecoles · France · centrale-maths2__pc Probability Generating Functions Explicit computation of a PGF or characteristic function

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

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