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The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. Moreover, the normal distribution is unique among distributions with finite variance in having this property. Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.

(The sort of obvious answer from teaching statistics several times:)

The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. 

Moreover, the normal distribution is unique among distributions with finite variance in having this property. 

Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.

The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. Moreover, the normal distribution is unique (among distributions with finite variance -- thanks Mark, for the comment below) in having this property. Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.

The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. Moreover, the normal distribution is unique among distributions with finite variance in having this property. Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.

The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. Moreover, the normal distribution is unique in having this property. Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.

The sum of two independent normal random variables is again normal, i.e., the shape of the distribution is unchanged under addition except for stretching and scaling. Moreover, the normal distribution is unique (among distributions with finite variance -- thanks Mark, for the comment below) in having this property. Many phenomena in nature come from adding together various independent or almost independent terms. Therefore, we would expect the normal distribution to show up a lot in nature-inspired mathematics.