(a) For any random variable $X$ and $\lambda>0$ and $t>0$, show that

$$\mathbb{P}(X>t) \leqslant \mathbb{E}\left(e^{\lambda X}\right) e^{-\lambda t}$$

For a standard normal random variable $X$, compute $\mathbb{E}\left(e^{\lambda X}\right)$ and deduce that

$$\mathbb{P}(X>t) \leqslant e^{-\frac{1}{2} t^{2}}$$

(b) Let $\mu, \lambda>0, \mu \neq \lambda$. For independent random variables $X$ and $Y$ with distributions $\operatorname{Exp}(\lambda)$ and $\operatorname{Exp}(\mu)$, respectively, compute the probability density functions of $X+Y$ and $\min \{X, Y\}$.