a) Compute the mean value

$$
\mathbb{E}\left[\operatorname{Max}_{t \in[0, T]} \widetilde{W}_{t}ight]=\mathbb{E}\left[\operatorname{Max}_{t \in[0, T]}\left(\sigma W_{t}+\mu tight)ight]
$$

of the maximum of drifted Brownian motion $\widetilde{W}_{t}=W_{t}+\mu t$ over $t \in[0, T]$, for $\sigma>0$ and $\mu \in \mathbb{R}$. The probability density function of the maximum is given in (10.14).

b) Compute the mean value $\mathbb{E}\left[\min _{t \in[0, T]} \widetilde{W}_{t}ight]=\mathbb{E}\left[\min _{t \in[0, T]}\left(\sigma W_{t}+\mu tight)ight]$ of the minimum of drifted Brownian motion $\widetilde{W}_{t}=\sigma W_{t}+\mu t$ over $t \in[0, T]$, for $\sigma>0$ and $\mu \in \mathbb{R}$. The probability density function of the minimum is given in (10.17).