Why do we need weighted distributions? I have read some papers about Weighted Distribution. Suppose $X$ is a non-negative continuous random variable with pdf $f(x)$. The pdf of the weighted random variable $X_w$ is given by:
$f_w (x) = \frac{w(x)f(x)}{E[w(X)]}$, where $w(x)$ is a non-negative weight function.
My question is: why do we need to add weight function $w(x)$ to the $f(x)$? I mean, what is the idea behind the weighted distribution? What are the effects after adding the $w(x)$ to the $f(x)$? I still can't find the answer that I want.

You can see this as a way of constructing new distributions from old ones. Disregarding your requirement that $X\ge 0$, let $X$ be a standard normal variable, $\phi(x)$ its density and $\Phi(x)$ its cdf. Then we have with
$$w(x)=2\Phi(\alpha x)$$ that your construction becomes the skew-normal density.
You can replace the normal density above with other densities, and get other examples.
