Combining probability distributions

This table is yet unfinished.

If I observe the sum of two processes with known distributions, the distribution of the observations is expected to be …

 + Normal Poisson Binomial Uniform
Normal Normal See here
Poisson Poisson,
normal if many summands
Binomial Binomial, if common p,
Poisson, if many summands,
Poisson binomial otherwise, note also the binomial sum variance inequality and this.
Uniform Irwin-Hall

Wikipedia has some more. A general discussion of probability distribution convolutions is for example here.

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Please simplify: \(\displaystyle\sum_{i=1}^{7}i=\)