Normal vs heavy tail

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats

plt.close('all')

mu = 1461     # taken from Bilalic et al.: "Why are (the best) women so good at chess? Participation rates and gender differences in intellectual domains", fig. 1
sigma = 342

nu = 10        # degrees of freedom for t-distribution

G = scipy.stats.norm(mu, sigma)     # normal distribution
T = scipy.stats.t(nu, mu, sigma)     # t-distribution

x = np.linspace(0,3000,1000)

plt.figure()
plt.plot(x,G.pdf(x))
plt.plot(x,T.pdf(x))
plt.xlabel('ELO rating')
plt.ylabel('likelihood')
plt.legend(['Gaussian distribution', 't-distribution '+str(nu)+' dof']); plt.grid()

# survival function of a 3.5 sigma event
p_eg = G.sf(mu+3.5*sigma)     # probability of event when gaussian
p_eng = T.sf(mu+3.5*sigma)    # probability of event when not gaussian

p_g = np.array([0.5,0.7,0.8,0.9,0.95,0.98,1])    # prior probabilites for Gaussianity

result = (p_g*p_eg) / (p_g*p_eg+(1-p_g)*p_eng)

plt.figure()
plt.plot(p_g,result,'*-')
plt.plot([0.5,1],[0.5,0.5],'r--')
plt.xlabel('prior probability that data is Gaussian')
plt.ylabel('P(Gaussian | 3.5 sigma event)')
plt.title('Probability that data is Gaussian conditional on observing a 3.5 sigma event\n(eqivalent to an ELO-rating of 2685))'); plt.grid()