Ch03 sum of two Gaussians is Gaussian?

[wilcobonestroo]

In chapter 3 there is a sentence

A remarkable property of Gaussian distributions is that the sum of two independent Gaussians is another Gaussian

To me, this looks like an error. For example, if they have different means, the sum will be a function with two "bumps". Is this correct, or am I missing something here?

[rlabbe]

This is poorly written. The sum of two independent, normally distributed variables is proportional to another gaussian. I thought I had all of these mis-expressions cleaned up, but apparently not.

https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

[wilcobonestroo]

Hmm... My confusion was in the distinction between the distribution (function) and random variable. The sum of the two functions is not a normally distributed function, but the sum of two random variables that are normally distributed is normally distributed. Wikipedia describes it as:

Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed

Edit: I had not refreshed my browser, and I wrote my message before I read your response. It is indeed about the random variables.

[KumarRishabh]

In my understanding, the concept that generally confuses people is that of convolution of random variables vs mixture of distributions. To avoid this confusion, extensive treatment of the topic might be required