Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria:
$$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad \int\limits_0^{+\infty}f_1(x)\mathrm{d}x = 1$$
What happens to this probability density if we censure all values $x < x_0$ to $x_0$ (such as we replace all data less than a Limit of Detection by this threshold).
I though that it might be something like:
$$f_2(x) = \left\{ \begin{matrix} 0,& 0 \leq x < x_0 \\ a\cdot\delta(x-x_0),& x=x_0 \\ f_1(x),& x > x_0& \end{matrix} \right.$$
Where $\delta(x)$ is the Dirac function and $a$ an arbitraty constant such as $a = \int\limits_0^{x_0}f_1(x)\mathrm{d}x$ in order to keep the area of $f_2$ unitary.
My questions are the following:
- Is this development valid?
- If so, what caution must be taken when dealing with discontinuous probability density?
- Is there a good reference about this kind of problem?