Question

Let X be a random variable on (0, 1) whose density is f(x). Show that we can estimate # 1 0 g(x) dx by simulating X and then taking g(X)/f(X) as our estimate. This method, called importance sampling, tries to choose f similar in shape to g, so that g(X)/f(X) has a small variance.

Short answer

The required statement is proved below.

Step-by-step solution

Given Information

We need to prove the statement.

Explanation

Consider$\frac{g\left(X\right)}{f\left(X\right)}$is a random variable, as the function of a random variable $X$. Using the theorem regarding the expectation of the function of the random variable, we have

$E\left[\frac{g\left(X\right)}{f\left(X\right)}\right]={\int }_0^1\frac{g\left(x\right)}{f\left(x\right)}·f\left(x\right)dx={\int }_0^{1}g\left(x\right)dx$

Since, we have the expected value of $\frac{g\left(X\right)}{f\left(X\right)}$is equal to the required value ${\int }_0^{1}g\left(x\right)dx$, we have that it is a good estimater of $\frac{g\left(x\right)}{f\left(x\right)}$. In statistics, these estimators that have the mean equal to the estimated value are called unbiased estimators.