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Which equation does Matlab use to fit a Normal distribution?

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Which equation does Matlab use to fit a Normal distribution?, especially for fitdist(x,'Normal') function. I would like to know if the fitting is given by:
or by another expression.
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Andreina Urquiola
Andreina Urquiola 11 Feb 2021
Different expressions can be used to minimize errors. For example Origin (data analysis software) uses this equation to obtain the gaussian fit:
I would like to know which expression is used by Matlab.

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Matt J
Matt J 11 Feb 2021
편집: Matt J 11 Feb 2021
To reverse engineer, we can fit the following data
x=randn(1e5,1)*2+5;
which is definitely distributed according to the first density parametrization that you posted (what I would consider the classical parametrization) with mu=5 and sigma=2. Since the routine gives us virtually the same sigma and mu as what we simulated, there is a good chance that the parametric model is the classical one:
pd = fitdist(x,'Normal')
pd =
NormalDistribution Normal distribution mu = 4.99612 [4.98373, 5.00851] sigma = 1.99916 [1.99044, 2.00796]
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Matt J
Matt J 12 Feb 2021
You're welcome, but please Accept-click an answer that you deem best solves your question.

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John D'Errico
John D'Errico 11 Feb 2021
I think you misunderstand, as there are several things happening here.
A normal distribution fit, that is, finding the parameters of a Normal PDF assumes the distribution has the property that the integral of that function is 1. A PDF has that property, and this is implicit in tools like fitdist.
However, IF you use some other software to perform a nonlinear regression fit, to a model as you show, thiis is NOT a normal distribution. It fails the property that the integral of that function is 1. In fact, the integral from -inf to inf is unbounded, if y0 is ANY number other than zero.
So it is NOT a normal distribution. You may call it a Gaussian fit, but the only thing it has in common with a normal distribution is it looks like a normal, and it is based on the same equation, though perhaps with some additional parameters. It is NOT a normal distribution.

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