Graphical histogram with a theoretical curve: random implementation

Question

I need to write a program that generates random implementations of the Cauchy distribution

with zero location and units scale.

I also need to make a histogram between -5 and 5 cells, for a random implementation of 1000 points, together with a theoretical curve , to make sure that they have the same units.

I calculated the cumulative distribution function for the Cauchy distribution:

And I wrote the following python code:

from __future__ import division
import scipy
import random
import matplotlib.pyplot as plt
import numpy as np
import math as m


valuesX = []
for q in range(1000):
    R = random.random()
    x = m.tan(m.pi*(R-0.5)) #Cumulative Function
    valuesX.append(x)

z = np.linspace(-10,10,1000)
y = 1/(m.pi*(1+z**2)) #Theoretical Cauchy

plt.plot(y,z)
plt.hist(valuesX, bins = 50, range = [-5,5], normed=True)

My results:

, , ( ) . ? .

Answer 1

, , , , . "normed = True" plt.hist() . .

, , , , , , .

, , [a, b], :

, .

, :

, (.. b ). , .