# Sum of independent random variables is $\sim\mathrm{Exp}(\lambda)$

Consider a parameter $$\lambda > 0$$ and a random variable $$Y$$ taking values in $$\mathbb{N}$$, with the support of its distribution being $$\mathbb{N}$$. Given that $$Y=n$$ for some $$n\in\mathbb{N}$$, is there a canonical way to construct independent random variables $$(X_i^{(n)})_{i=1}^n$$ such that $$\sum_{i=1}^n X_i^{(n)} \sim \mathrm{Exp}(\lambda)?$$

Note that $$\mathrm{Exp}(\lambda)$$ must not depend on $$n$$.

The exponential distribution is a special case of the Gamma distribution with $$\alpha=1,\,\beta=\lambda$$. In general its characteristic function is $$\left(1-\frac{it}{\beta}\right)^{-\alpha}$$. We can thus take iids $$X_i^{(n)}$$ to also be Gamma-distributed with $$\alpha=\frac{1}{n},\,\beta=\lambda$$.