Which distributions can be combined linearly, assuming they are independent? Let $X$ and $Y$ be two independent distributions. 
I want a list of the distributions that can be combined linearly.
For example, if $X\sim \text{Poisson}(\lambda)$ and 
$Y\sim \text{Poisson}(\lambda)$ and $Z= X + Y$, then $Z\sim \text{Poisson}(2\lambda)$
What are the distributions for which this is true? 
Useful Link:
http://www.randomservices.org/random/special/Divisible.html
 A: There is a 4-parameter family of these called stable distributions.  
The question states that distributions (of some type) can be "combined linearly". That condition excludes the binomial and Poisson distributions, which are only closed under taking sums of independent distributions from the family.  If $X$ is binomial or Poisson, then $2X$ is not.  
If you only want the sum of independent distributions and not general linear combinations, there are non-stable (in fact non-divisible) distribution families closed under independent sum as explained in Michael Hardy's answer. 
A: Here you're looking not for individual distributions, but for families of distributions.  Suppose $X$ is a real-valued random variable with any distribution at all and $X_1,X_2,X_3,\ldots$ are independent copies of $X$.  Then the distributions of $X_1$, $X_1+X_2$, $X_1+X_2+X_3$, has the property in question, so it's quite a large class of families of distributions.
Such a family of distributions need not be infinitely divisible.  For example suppose consider the family of binomial distributions with parameters $n$ and $p$, for $n\in\{0,1,2,\ldots\}$ for fixed $p$.  That has just the property you've specified but is not infinitely divisible.
Nor need they be stable distributions: the Poisson and binomial families are not stable.  (Stable distributions are those for which the distributions of $X_1,\   X_1+X_2,\  X_1+X_2+X_3,\  \ldots$ differ only in location and scale.  The Poisson and binomial families are not stable in that sense.
