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:
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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