grandes-ecoles 2018 Q12

grandes-ecoles · France · centrale-maths2__psi Moment generating functions Compute MGF or characteristic function for a named distribution

For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.
Calculate the moment generating function of the random variable $Y _ { n }$.

For $n \in \mathbb { N } ^ { * }$, $U _ { n }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following the uniform distribution on $\llbracket 1 , n \rrbracket$. We set $Y _ { n } = \frac { 1 } { n } U _ { n }$.

Calculate the moment generating function of the random variable $Y _ { n }$.

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