# Origin: Skbkekas
# Enhanced: Ika, 2013-07-24

import numpy as np
import matplotlib.pyplot as plt
import scipy.special as sp
 
col = ['orange', 'purple', 'deepskyblue']
 
X = np.arange(-5, 5, 0.001)
 
plt.clf()
plt.figure(figsize=(4,3.2))
plt.axes([0.17,0.13,0.79,0.8])
plt.hold(True)
 
A = []
for k,nu in enumerate([1,2,5]):
    Y = -(nu+1)*np.log(1+X**2/nu)/2
    Y += sp.gammaln((nu+1)/2.0)
    Y -= sp.gammaln(nu/2.0)
    Y -= 0.5*np.log(nu*np.pi)
    a = plt.plot(X, np.exp(Y), '-', color=col[k], lw=1.5)
    A.append(a[0])

# Draw the curve of Normal distribution, which is the limit of t-distribution sequence.
mu = 0	# mean = 0
sigma = 1	# variance = 1
M = 1/(sigma*np.sqrt(2*np.pi))
N = np.exp(-(X-mu)*(X-mu)/(2*sigma*sigma));
Y = M*N
a = plt.plot(X, Y, '-', color='black', lw=1.5)
A.append(a[0])

plt.xlabel("x")
plt.ylabel("P(x)")

bx = plt.legend(A, (r"$\nu=1$", r"$\nu=2$", r"$\nu=5$", r"$\nu=+\infty$"),\
                numpoints=1, handlelen=0.05, handletextpad=0.4,\
                loc="upper right")
bx.draw_frame(False)
plt.xlim(-5,5)
 
plt.savefig("student_t_pdf.pdf")
plt.savefig("student_t_pdf.svg")