Monte Carlo Methods

Algorithm Monte Carlo Integration
Given a sample $x(1),\dots ,x(N)\sim f(x)$, we gan approximate the theoretical mean by the sample mean to give: $E_f[\varphi (X)]=\int \varphi (x)f(x)\text{dd}x\approx \frac{1}{N}{\sum }_{i=1}^N\varphi \left(x^{(i)}\right)$

Thrm By Monte Carlo Integration, we can derive the following approximation formulas: $\begin{array}{r}{\int }_a^{b}g(x)\text{dd}x={\int }_0^1(b-a)g(a+u(b-a))\text{dd}u\\ {\int }_0^{\infty }g(x)\text{dd}x={\int }_0^1\frac{1}{u^2}g\left(\frac{1}{u}-1\right)\text{dd}u\end{array}$
e.g. ${\int }_0^2\exp (x^2)\text{dd}x=2{\int }_0^1\exp (4u^2)\text{dd}u$, thus we can estimate the integral by the following R code:

u=runif(5000)
mean(2*exp(4*u^2))