https://github.com/cuadernin/integralsimulation
Computes the value of a definite integral using Monte Carlo simulation.
https://github.com/cuadernin/integralsimulation
integral monte-carlo monte-carlo-simulation simulation
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Computes the value of a definite integral using Monte Carlo simulation.
- Host: GitHub
- URL: https://github.com/cuadernin/integralsimulation
- Owner: Cuadernin
- Created: 2021-02-04T03:46:14.000Z (over 5 years ago)
- Default Branch: master
- Last Pushed: 2021-06-19T03:57:33.000Z (about 5 years ago)
- Last Synced: 2024-05-17T16:59:19.998Z (about 2 years ago)
- Topics: integral, monte-carlo, monte-carlo-simulation, simulation
- Language: Python
- Homepage:
- Size: 11.7 KB
- Stars: 1
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# 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 fx and g is a continuous function, the expectation of g(X) is given by:
)=\int_{-\infty&space;}^{\infty&space;}g(x)f_x(x)dx{\color{Blue}&space;} "\LARGE E(g(X))=\int_{-\infty }^{\infty }g(x)f_x(x)dx{\color{Blue} }")
If X is uniformly distributed in [0, 1], then
)=\int_{0}^{1}g(x)dx "\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, ..., Xn are independent and uniformly distributed in [0, 1], then the g(Xi) are also independent.
Therefore, if we define the mean of g(X) as:
}=\frac{g(X_1)+g(X_2)+...+g(X_n)}{n} "\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:
})=\mu&space;\phantom{abcd}&space;V(\overline{g(X)})=\frac{\sigma^2}{n} "\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(un))/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
## Output 📦
* **Simulated value**
* **Real value**
* **Percentage error**
### Note
You can do the exact same thing for a multiple integral.