Using simulation, I have estimated the probability density function ($F$) of a random variable ($X$). Now I measure the value of one instance ($x$) and the standard deviation of it. The uncertainty of the measurement is approximated with a normal distribution ($N$). If I'm not wrong, the probability that $x$ is between $a$ to $b$ is $\int_a^bNF \;dx$. This should be a very common procedure. But I cant remember anywhere I have seen something like this example, thus don't know what keywords I should search for.
Excessive googling with anything related I could think of didn't help. It seems a silly question, but do know where I can find examples similar to this? or how this process is described or some pointers?