grandes-ecoles 2023 Q26

grandes-ecoles · France · centrale-maths2__official Normal Distribution Tail Bound or Concentration Inequality Proof
The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
For all $\ell > 0$, show that there exists a natural number $n _ { 2 }$, such that, for all $n \geqslant n _ { 2 }$, $$B _ { n } ( \ell ) \leqslant 2 \varphi ( \ell )$$ 
The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

For all $\ell > 0$, show that there exists a natural number $n _ { 2 }$, such that, for all $n \geqslant n _ { 2 }$,
$$B _ { n } ( \ell ) \leqslant 2 \varphi ( \ell )$$
Paper Questions
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