Let $\left(\Omega, \mathcal{F}, P\right)$ a probability space, and $X$ be an $N\left(0, \sigma^2\right)$-distributed real-valued random variable. For any $\theta \in \mathbb{R}$, compute
(a) $E\left(e^{\theta X}\right)$
(b) Prove that
$E\left(e^{\theta X^2}\right) = \begin{cases} \frac{1}{\sqrt{1 - 2\theta\sigma^2}} & \text{if } \theta\sigma^2 < \frac{1}{2} \\ +\infty & \text{if } \theta\sigma^2 \ge \frac{1}{2} \end{cases}$