compute_aggregated_density.py

import numpy as np

mu_A = np.log(10)
sigma_A = 0.1

mu_B = np.log(20)
sigma_B = 0.1

def lognorm_pdf(x,mu,sigma):
  return  np.exp(-(np.log(x)-mu)**2/(2*(sigma**2))) / (x*sigma*np.sqrt(2*np.pi))

f_A = lambda x : lognorm_pdf(x, mu_A, sigma_A)
f_B = lambda x : lognorm_pdf(x, mu_B, sigma_B)

import scipy.integrate as integrate
f_aux = lambda x : np.sqrt(f_A(x)) * np.sqrt(f_B(x))
denominator, error = integrate.quad(f_aux, 0.1, np.inf)

assert np.abs(denominator/100) > error, f"The error is too big! {denominator, error}"

f = lambda x : f_aux(x) / denominator
expected_value, error = integrate.quad(lambda x : x*f(x), 0, np.inf)
assert np.abs(expected_value/100) > error, f"The error is too big! {denominator, error}"

print(f"The expected value of the aggregated density is {expected_value}")

mean_A = np.exp(mu_A + sigma_A**2 / 2)
mean_B = np.exp(mu_B + sigma_B**2 / 2)
print(f"The mean of the individual distributions is {mean_A, mean_B}")

geo_mean = np.sqrt(mean_A*mean_B)
print(f"The geometric mean of the means of the individual distributions is {geo_mean}")