grandes-ecoles 2023 Q27

grandes-ecoles · France · centrale-maths2__official Central limit theorem
The function $B _ { n }$ is defined as in Q19, $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$, and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$.
Conclude that the sequence $\left( \Delta _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges to 0. 
The function $B _ { n }$ is defined as in Q19, $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$, and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$.

Conclude that the sequence $\left( \Delta _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges to 0.
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29 Q30 Q31 Q32 Q33