Integral_Calculate_with_MonteCarlo_Methods.md

Using MonteCarlo Method for Integrate

habib ezzatabadi (stats9)

First Method; MonteCarlo Simulation —-

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Suppose we want to calculate the following integral using the Monte Carlo method.

$$ \int_{0}^{+ \infty} \exp(-x)dx $$

For this purpose, we must first transfer this integral to a limited interval with a suitable transformation.

$$ \begin{aligned} & u = \frac{1}{x + 1} \implies \\ & x + 1 = \frac{1}{u} \implies x = \frac{1}{u} - 1 \implies \\ & dx = -\frac{1}{u^2}du \implies \int_{0}^{+\infty} \exp(-x)dx = \int_1^0 -\frac{1}{u^2}\times \exp{-(\frac{1}{u} - 1)}du \\ & = I = \int_0^1 \frac{1}{u^2}\times \exp{-(\frac{1}{u} - 1)}du \implies \\ & \text{if} \quad g(u) = \frac{1}{u^2}\times \exp{-(\frac{1}{u} - 1)} \implies \\ & I \approx \frac{1}{n} \sum_{1}^n g(u_i), \quad U_i \overset{iid}{\sim} \text{Uniform}(0, ~1), ~~ n = \text{number of generate} \implies \end{aligned} $$

Code Soloution in R:

set.seed(1)
f <- function(x) exp(-x)
g <- function(u) 1/u**2 * exp(-(1/u - 1)) # define g
n <- 1e+5 # number of generate
U <- runif(n) # generate uniform random variables 
I <- mean(g(U))

Exact_value <- integrate(f, 0, Inf)$value
result <- data.frame(Integral_Method = c("MonteCarlo", "Exact_Value"), 
            Value = c(I, Exact_value))
result |> 
        knitr:: kable(caption = "Results", align = "c")

Integral_Method	Value
MonteCarlo	0.9974524
Exact_Value	1.0000000

Results

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Method II

Importance Sampling Method

Suppose we want to take the integral of the following function.

$$ I = \int_0^\infty x^2 \exp(-x^2)dx $$

Soloution:

$$ \begin{aligned} & \text{if} \quad g = x^2 \times \exp(-x^2) = x \times x \times \exp(-x^2) \\ & \text{if}\quad U \sim Weibull(\alpha = 2, \lambda = 1) \implies \\ & f_U(u) = 2x \times \exp(-x^2) \\ & \implies I = \mathcal{E}(\frac{1}{2}U) = \frac{1}{2} \times\mathcal{E}(U) \end{aligned} $$

Code Soloution in R:

n <- 1e+6
W <- rweibull(n, shape = 2, scale = 1)
res <- 1/2 * mean(W)
g <- function(x) x**2 * exp(-x**2)

Exact_value <- integrate(g, 0, Inf)$value

result <- data.frame(Integral_Method = c("MonteCarlo", "Exact_Value"), 
            Value = c(res, Exact_value))
result |> 
        knitr:: kable(caption = "Results", align = "c")

Integral_Method	Value
MonteCarlo	0.4430934
Exact_Value	0.4431135

Results