By definition MGF of rv \(X\) is \begin{align*} M_X(t) &= \mathbb{E}[e^{tX}] \\ &=\mathbb{E}\left [{(e^t)}^{X}\right ] \tag{1} \end{align*} and PGF of rv \(X\) is \begin{align*} G_X(s) &= \mathbb{E}[s^{X}] \tag{2} \end{align*} By (1) and (2) we see that \begin{align*} G_X(e^t) &= \mathbb{E}\left [{(e^t)}^{X}\right ] \\ &= M_X(t) \end{align*}
