Expected value conditioned on subset of domain I have a random variable $X$ with pdf $f(\cdot)$ and support $[0, b]$. As part of some results I am generating, I am left with the integral $$\int_a^b x f(x) \, dx$$ where $0<a<b.$ I am wondering how to think about this integral - can this be expressed $E[X\mid X>a]$ or something like this? Seems like a straightforward question I know but I couldn’t think of the right search terms to use. I’d welcome a link to another post asking the same thing if it exists.

If $S$ is any subset of $(a,b),$ then you have
\begin{align}
& \Pr(X\in S\mid X>a) = \frac{\Pr(X\in S\ \&\ X>a)}{\Pr(X>a)} \\[8pt]
= {} & \frac{\Pr(X\in S)}{\Pr(X>a)} = \frac{\int_S f(x)\,dx}{\int_a^b f(x)\, dx}.
\end{align}
Therefore the conditional probability density of $X$ given that $X>a$ is
$$
f_{X\,\mid\,X\,>\,a}(x) = \frac{f(x)}{\int_a^b f(u)\,du}.
$$
So you integrate $x$ times that to get the conditional expected value.
