Integral using importance sampling formula

Question

I am trying to do an integral via importance sampling formula. The algorithm is as follows: Code

The value for the integral should give: 0.838932960013382

Also I must use the next probability distribution to generate 1,000,000 random numbers between 0 and 1. I also use the next weight function. Finally with these numbers I must calculate this formula. But I am getting the numerical value wrong, and I am not sure about the calculation for the 1.000.000 random numbers.

Answer 1

You don't compute weights normalization and inverse properly, check your math

Code below, Python 3.9, Win 10 x64

import numpy as np
from scipy import integrate

def f(x):
    return 1.0/((np.exp(x)+1.0)*np.sqrt(x))

def w(x):
    return 0.5/np.sqrt(x)

def inv(x):
    return x*x

rng = np.random.default_rng()

N = 100000

x = rng.random(N)

p = inv(x)

q = f(p)/w(p)

print(np.mean(q))

print(integrate.quad(f, 0, 1))

prints

0.8389948486429488
(0.8389329600133858, 2.0727863869751673e-13)

looks good?