The sum of Two Identical Normals is Normal
From www.norsemathology.org
Normal Density function
Normally distributed variable x with mean and standard deviation has density function
The Distribution of the Sum of Two iid Normal Variables
Consider the sum s of two of these random variables x. The density of s is given by the convolution of the densities of the two:















Conclusion: the sum is normally distributed, with mean , and with standard deviation .
More Generally
More generally, the sum of two normals is normal, with parameters mean
and variance
By induction, the sum of n normals will be normal, with parameters
and