I'm doing data analysis research and currently I'm trying to fit the simple model to a data sample. From physical speculations

I know that the distribution should be a sum of two normal distributions with different parameters (mean, width, area). I fitted

my data with the model and it works pretty well. For every gaussian I have it's mean, sigma and area and their errors

but I'm wondering how to calculate the parameters(and their errors) of the fitted sum? I was thinking about some kind of

weighted average with weights being the areas of respective functions but I couldn't find any mathematical proof for this.

Regards

M