Question

Explain how you could use random numbers to approximate ${\int }_0^{1}k\left(x\right)dx$, where k(x) is an arbitrary function.

Short answer

The answer of the question is${\int }_0^{1}k\left(u\right)du=E\left(k\right(U\left)\right)=\frac{1}{n}\sum _{i=1}^{n}k\left(U_i\right)$

Step-by-step solution

Given Information

We need to explain the use of random numbers to approximate${\int }_0^{1}k\left(x\right)dx$

Simplify

Consider $U~Unif(0,1).$Using the theorem about the expectation of the function of the random variable, we have that

$E\left(k\right(u\left)\right)={\int }_{\mathrm{ℝ}}k\left(u\right)f_U\left(u\right)du={\int }_0^{1}k\left(u\right)du$

On the other hand, we can estimate the mean of random variable with the mean of the sample. So, taking the independent variable $U_1,...,U_n~U$and considering its functional values localid="1648210961949" $k\left(U_1\right),...,k\left(U_n\right)$.Therefore, we can estimate

$E\left(k\right(U\left)\right)=\frac{1}{n}\underset{i=1}{\overset{n}{\sum k(U_i}})$