How to fit data to a normal distribution using MLE and Python

MLE, distribution fittings and model calibrating are for sure fascinating topics. Furthermore, from the outside, they might appear to be rocket science. As far I'm concerned, when I did not know what MLE was and what you actually do when trying to fit data to a distribution, all these tecniques did looked exactly like rocket science.
They are not that much complicated though. MLE is a technique that enables you to estimate the parameters of a certain random variable given only a sample by generating a distribution which makes the observed results the most likely to have occurred. Distribution fittings, as far as I know, is the process of actually calibrating the parameters to fit the distribution to a series of observed data.

Let's see an example of MLE and distribution fittings with Python. You need to have installed scipy, numpy and matplotlib in order to perform this although I believe this is not the only way possible. For some reason that I ignore, the methods in scipy.stats related to the normal distribution use loc to indicate the mean and scale to indicate the standard deviation. I maybe can grasp why use "scale" to indicate the stdv however I really do not get "loc" I do not understand why... If you know that, please leave a comment.

	from scipy.stats import norm
	import matplotlib.pyplot as plt
	import numpy as np
	# Generate an array of 200 random sample from a normal dist with
	# mean 0 and stdv 1
	random_sample = norm.rvs(loc=0,scale=1,size=200)
	# Distribution fitting
	# norm.fit(data) returns a list of two parameters
	# (mean, parameters[0] and std, parameters[1]) via a MLE approach
	# to data, which should be in array form.
	parameters = norm.fit(random_sample)
	# now, parameters[0] and parameters[1] are the mean and
	# the standard deviation of the fitted distribution
	x = np.linspace(-5,5,100)
	# Generate the pdf (fitted distribution)
	fitted_pdf = norm.pdf(x,loc = parameters[0],scale = parameters[1])
	# Generate the pdf (normal distribution non fitted)
	normal_pdf = norm.pdf(x)
	# Type help(plot) for a ton of information on pyplot
	plt.plot(x,fitted_pdf,"red",label="Fitted normal dist",linestyle="dashed", linewidth=2)
	plt.plot(x,normal_pdf,"blue",label="Normal dist", linewidth=2)
	plt.hist(random_sample,normed=1,color="cyan",alpha=.3) #alpha, from 0 (transparent) to 1 (opaque)
	plt.title("Normal distribution fitting")
	# insert a legend in the plot (using label)
	plt.legend()
	# we finally show our work
	plt.show()

view raw fit_normal_dist.py hosted with ❤ by GitHub

The result should look somewhat like this:

Hope this was useful.