<jupyter_start><jupyter_text>Set up a series of random uniform distribtion<jupyter_code>n_exp= 10000
n_draw = 10
n_bins = 100
x_min =0.15
x_max = 0.85

means = np.zeros(n_exp)<jupyter_output><empty_output><jupyter_text>Perfrom the experiment<jupyter_code>for i in range(n_exp):
    z = np.random.uniform(0.,1.,n_draw)
    
    means[i] = np.mean(z)
def gaussian(x,mu,sigma):
    return 1./(2.0*np.pi*sigma**2)**0.5 * np.exp(-0.5*((x-mu)/sigma)**2)<jupyter_output><empty_output><jupyter_text>Plot Histogram<jupyter_code>fig = plt.figure(figsize=(7,7))

y_hist, x_hist, ignored = plt.hist(means,bins = n_bins, range = [x_min,x_max], density=True)

xx = np.linspace(x_min,x_max, 1000)

sigma= 1./12.0 **0.5 / 10.0**0.5

plt.plot(xx, gaussian(xx,0.5,sigma),color='red')
#plt.plot([0,1.1*gaussian(0.5,0.5,sigma)])
plt.xlim(x_min,x_max)

plt.xlabel('x')
plt.ylabel('y(x)')

plt.show()<jupyter_output><empty_output>
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