Sum of two normal distributions


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.

A normal distribution can be represented as a sum of infinitely many normal distributions, and in your case just two. This is easy to see/prove when you use moment generating functions. Lets assume Z is your observed data, then you can write it as Z = X + Y. Where the mean of Z needs to be equal to the mean of X + mean of Y. Also, the variance of Z is equal to the Var(X)+Var(Y).

Not sure if this helps, but do explain more so we can help.


Ambassador to the humans
Also a normal distribution only takes two parameters but you're talking about three. What are those all representing?


Ambassador to the humans
You're assuming independence there. You need to account for the covariance otherwise. But honestly we typically care about more than just the mean and variance.
Thank You for all replies. I think I wasn't clear with my question and we're talking about two different things. What You refer to is the sum of two (or more) normally distributed random variables. What I am using is a sum (mixture) of two normal distributions, which I use to fit the model with. So the value at x is something like f(x) = A1*f1(x) + A2*f2(x), with A_i being the weight which is basically the height of Gaussian at mean value (cause they're not necessarily normalised). From the fit I got A_i, mu_i, sigma_i and their errors and I want to find the mean of the mixture. Thanks in advance and sorry for late response!


Currently I estimated the mean as:

mu = (A1*mu_1 + A2*mu_2)/(A1+A2),

and calculated it's error from propagation theorem. I'm not sure if it's mathematically justified

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Ambassador to the humans
Shouldn't the weights (The Ai terms) sum to 1 if it's a proper mixture? Your formula for the mean if the mixture is accurate then but the denominator should reduce to 1.


Ambassador to the humans
I think some of the details are getting mucked up and it is hard to talk about the technical aspects if we aren't on the same page. Is there a reason you aren't using the canonical way to parameterize a gaussian mixture and instead are using your different offshoot?
You're right. I might be not very clear.
Canonical way to parametrize a mixture, with respective weights summing up to one would give me a pdf in result. I don't want to have that.
Instead I want to have a gaussian-like (or "gaussian mixture like") distribution that is normalized to a certain number (number of occurences), so the
y = f(x) represents the number of occurences for given x, and not the probability. To do it I parametrize a Gaussian with three parameters:
height, mean and sigma:
f_i(x) A_i * N(mean, sigma^2), where N(mean, sigma^2) is the pdf with given mean and sigma.

I added an example in the attachment, where there's cauchy + quadratic background. In my case it's two gaussians.