Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ We denote $G_{X}(t) = \mathrm{E}\left(t^{X}\right) = \sum_{k=0}^{\infty} \mathrm{P}(X = k) t^{k}$ (generating series of the random variable $X$). Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Determine $G_{X}(t)$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$:
$$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
We denote $G_{X}(t) = \mathrm{E}\left(t^{X}\right) = \sum_{k=0}^{\infty} \mathrm{P}(X = k) t^{k}$ (generating series of the random variable $X$).
Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Determine $G_{X}(t)$.