Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$: $$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$ Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Let $\mu$ be a strictly positive real number. Let $Y$ be a random variable following the Poisson distribution $\mathcal{P}(\mu)$ and such that $X$ and $Y$ are independent. Determine the distribution of $X + Y$.
Recall that a random variable $X$, taking values in $\mathbb{N}$, follows the Poisson distribution $\mathcal{P}(\lambda)$ with parameter $\lambda$ if, for all $n \in \mathbb{N}$:
$$\mathrm{P}(X = n) = \frac{\lambda^{n}}{n!} \mathrm{e}^{-\lambda}$$
Let $X$ be a random variable that follows the Poisson distribution $\mathcal{P}(\lambda)$. Let $\mu$ be a strictly positive real number. Let $Y$ be a random variable following the Poisson distribution $\mathcal{P}(\mu)$ and such that $X$ and $Y$ are independent. Determine the distribution of $X + Y$.