Question

Is the sum of two Gaussian functions still a Gaussian function? What implication does this have for adding data sets?

Answer 1

The sum of two normally distributed independent random variables will also be normally distributed.

Explanation:

Let's assume the question is asking about the sum of two random variables which each have Gaussian (or normal) probability density functions. The answer is that if the variables are independent, this is true. If we have two normally distributed random variables, `X` and `Y` where

`X => N(mu_x,sigma_x^2)`

and

`Y => N(mu_y,sigma_y^2)`

we define

`Z=X+Y`

then

`Z=>N(mu_x+mu_y,sigma_x^2+sigma_y^2)`

What this is saying is that the mean of the resulting normal distribution is the sum of the two source distributions, and the variance is the sum of the two variances.

When dealing with datasets which are normally distributed, this implies that the resulting sum will also be normally distributed if the points are independent samples.

This property can also result in the central limit theorem when we take the sum and then divide by the number of elements, i.e. take the average of the points:

`M = sum_i^N(X_i)/N`

such that

`M => N(mu_x,sigma_x^2/N)`