Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ Show that the function $m$ is strictly increasing on $\mathbb { R } _ { + }$, and that for all $t \in [ 0,1 [$, there exists a unique $\lambda \geqslant 0$ such that $m ( \lambda ) = t$.
Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$,
$$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$
For each $\lambda \geqslant 0$, we set
$$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$
Show that the function $m$ is strictly increasing on $\mathbb { R } _ { + }$, and that for all $t \in [ 0,1 [$, there exists a unique $\lambda \geqslant 0$ such that $m ( \lambda ) = t$.