Let \(Z\) be a \(\mathrm{N}(0,1)\) random variable and define \(X=\mu+\sigma Z\) for some \(\mu \in \mathbb{R}\) and \(0<\sigma<\infty\). Using the fact that \(X\) is a \(\mathrm{N}\left(\mu, \sigma^{2}ight)\) random variable, derive the moment generating function and the characteristic function of a \(\mathrm{N}\left(\mu, \sigma^{2}ight)\) random variable.