Assume $x$ follows a normal distribution $f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$. Therefore, if I want to build a simulation to characterize $x$, then I can just sample from $f(x)$. Now a new problem is, at one particular sampling, if a prior information is known that $x\geq 10$, then what distribution should I sample from?
I think it is a conditional probability derived from $f(x)$ and prior $x\geq 10$, but I am confused what exactly it is.
Let $F_X$ be the distribution function of $X$. For $x\geqslant 10$ we have by definition of conditional probability \begin{align} \mathbb P(X\leqslant x\mid X>10) &= \frac{\mathbb P(X\leqslant x,X>10)}{\mathbb P(X>10)}\\ &= \frac{F_X(x) - F_X(10)}{1-F_X(10)}\\ &= \frac{\text{erf}\left(\frac{\mu -10}{\sqrt{2} \sigma }\right)-\text{erf}\left(\frac{\mu -x}{\sqrt{2} \sigma }\right)}{\text{erf}\left(\frac{\mu -10}{\sqrt{2} \sigma }\right)+1}. \end{align}