Integral Simulation

Computes the value of a definite integral using Monte Carlo simulation in Python.

Problem to solve ✔

If X is a continuous random variable with density function f_x and g is a continuous function, the expectation of g(X) is given by:

$\LARGE E(g(X))=\int_{-\infty }^{\infty }g(x)f_x(x)dx{\color{Blue} }$

If X is uniformly distributed in [0, 1], then

$\LARGE E(g(X))=\int_{0}^{1}g(x)dx$

This fact suggests that we can use simulation to estimate the previous integral, estimating the value of E(g(X)). If X₁, X₂, X₃, ..., X_n are independent and uniformly distributed in [0, 1], then the g(X_i) are also independent. Therefore, if we define the mean of g(X) as:

$\LARGE \overline{g(x)}=\frac{g(X_1)+g(X_2)+...+g(X_n)}{n}$

and we assume that g(X) has mean μ and variance σ², then:

$\LARGE E(\overline{g(X)})=\mu \phantom{abcd} V(\overline{g(X)})=\frac{\sigma^2}{n}$

These two equalities suggest that to estimate μ, we can generate a sequence of numbers u₁, u₂, u₃, ..., a uniformly distributed in [0, 1], independent and calculate (g(u₁)+g(u₂)+g(u₃)+...+g(u_n))/n. Note that the variance of the mean g(X) can be made as small as you like by taking it large enough.

Input📋

A menu of options is displayed where you must choose the type of interval where the integral will be calculated, that is:

1 == [a, b]
2 == [0, 00]
3 == [-00 , 00]

Next, you must write the function in terms of x and the limits of the interval.

exp(-x**2)

It means $\LARGE e^{-x^2}$

Output 📦

Simulated value
Real value
Percentage error

Note

You can do the exact same thing for a multiple integral.

Cuadernin/IntegralSimulation

Integral Simulation

Problem to solve ✔

Input📋

Output 📦

Note