grandes-ecoles 2019 Q6

grandes-ecoles · France · centrale-maths2__mp Discrete Random Variables Expectation of a Function of a Discrete Random Variable

Using the result that $X_n$ and $-X_n$ have the same distribution, deduce the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$

Using the result that $X_n$ and $-X_n$ have the same distribution, deduce the pointwise limit of the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by
$$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$

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