grandes-ecoles 2020 Q19

grandes-ecoles · France · centrale-maths2__pc Continuous Probability Distributions and Random Variables Almost Sure Convergence and Random Series Properties

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$, and $\tilde{g}_n$ denotes its continuous extension to $\mathbb{R}^+$. Show that the function $G = \sum _ { n = 0 } ^ { + \infty } \tilde { g } _ { n }$ is defined and continuous on $\mathbb { R } ^ { + }$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$, and $\tilde{g}_n$ denotes its continuous extension to $\mathbb{R}^+$.\\
Show that the function $G = \sum _ { n = 0 } ^ { + \infty } \tilde { g } _ { n }$ is defined and continuous on $\mathbb { R } ^ { + }$.

Paper Questions

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