Show that the distribution of the random variable $X_n$ is given by $$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$
Show that the distribution of the random variable $X_n$ is given by
$$\forall k \in \llbracket 0, n \rrbracket \quad P_n\left(X_n = k\right) = \frac{1}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$