Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$ By applying Markov's inequality to a suitably chosen random variable, prove that $$\forall t \in \mathbb { R } ^ { + } , \quad \mathbb { P } ( Z \geqslant \lambda ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } - \lambda t \right)$$
Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies
$$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$
By applying Markov's inequality to a suitably chosen random variable, prove that
$$\forall t \in \mathbb { R } ^ { + } , \quad \mathbb { P } ( Z \geqslant \lambda ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } - \lambda t \right)$$