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I am looking for a finitely parameterized family of non-atomic distributions $D(\vec{\lambda})$, $\vec{\lambda} \in \mathbb{R}^k$ for some finite $k$, such that if $X \sim D(\vec{\lambda}_1)$ and $Y \sim D(\vec{\lambda}_2)$, then we have $X + Y \sim D(\vec{\lambda}_3)$ and $\max\{X,Y\} \sim D(\vec{\lambda}_4)$. Of special interest to me is when the distribution is supported on the $\mathbb{R}_+$, so that it can model the execution time of some process.

I have the following no-brainer way of constructing a family of distributions closed under taking maximum: Let $F:\mathbb{R}_+ \to [0,1)$ be an increasing surjective function. Define $F_\lambda = F^\lambda$, for $\lambda > 0$. Then clearly this gives the cdf for a family of such distributions. For instance we can take $F(t) = 1 - e^{-t}$, the exponential distribution with parameter $1$.

Now given the above family, to further require additive closure amounts to the following statement: let $f = F'$, and $\lambda, \mu > 0$, then we need $f_\lambda \ast f_\mu = f_\gamma$ for some $\gamma > 0$. Can this infinite set (parametrized by $\lambda,\mu$) of functional equations be solved? Alternatively I would be happy with $\lambda,\mu \in \mathbb{N}$, provided $\gamma \in \mathbb{N}$ also. The solution may require $\lambda,\mu,\gamma$ to be higher dimensional.

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