The sum of Two Identical Normals is Normal

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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