distributions.py

import random, pylab

#set line width
pylab.rcParams['lines.linewidth'] = 4
#set font size for titles
pylab.rcParams['axes.titlesize'] = 20
#set font size for labels on axes
pylab.rcParams['axes.labelsize'] = 20
#set size of numbers on x-axis
pylab.rcParams['xtick.labelsize'] = 16
#set size of numbers on y-axis
pylab.rcParams['ytick.labelsize'] = 16
#set size of ticks on x-axis
pylab.rcParams['xtick.major.size'] = 7
#set size of ticks on y-axis
pylab.rcParams['ytick.major.size'] = 7
#set size of markers, e.g., circles representing points
#set numpoints for legend
pylab.rcParams['legend.numpoints'] = 1

dist = []
for i in range(100000):
    dist.append(random.gauss(0, 30))
pylab.hist(dist, 30)
#pylab.show()

import scipy.integrate

def gaussian(x, mu, sigma):
    factor1 = (1.0/(sigma*((2*pylab.pi)**0.5)))
    factor2 = pylab.e**-(((x-mu)**2)/(2*sigma**2))
    return factor1*factor2

def checkEmpirical(numTrials):
  for t in range(numTrials):
     mu = random.randint(-10, 10)
     sigma = random.randint(1, 10)
     print('For mu =', mu, 'and sigma =', sigma)
     for numStd in (1, 1.96, 3):
        area = scipy.integrate.quad(gaussian,
                                    mu-numStd*sigma,
                                    mu+numStd*sigma,
                                    (mu, sigma))[0]
        print('  Fraction within', numStd, 'std =', round(area, 4))

checkEmpirical(3)