# Divisibility of the exponential distribution

Let \(Z\) be a \(\text{Exp}(1)\) random variable. For \(\alpha_1, \ldots \alpha_N \in \mathbb{R}_{+}^{*}\), we are looking for variables \(X_1, \ldots X_N\) independent and following the same distribution such that:

\[Z = \sum_{j=1}^{N} \alpha_j X_j.\]For the special case where all \(\alpha_j\) are equal, this corresponds to the problem of *infinite divisibility* of the exponential distribution, and in this case \(X_1\) follows \(\text{Gamma}(1/N, 1/\alpha_1)\), a gamma variable with shape \(1/N\) and scale \(1/\alpha_1\) (in particular when \(N=1\), we have \(X_1 \sim \text{Exp}(\alpha_1) = \text{Gamma}(1, 1/\alpha_1)\)).

We are willing to extend this divisibility property by deriving the characteristic function of \(X_1\) before inversing it numerically to retrieve the density. Some visualizations are provided.