Let $X$ be a non negative real valued random variable with say continuous pdf $f$. Is the function $g(b)=\mathbb{E}(b+X\mid b+X \geq a)$ for $b \leq a$ increasing?
It seems intuitively true to me but so far I didn't manage to prove it. It seems to be a similar question as Conditional Expectation of Increasing Function.
My approach was to regard the derivative of
$$g(b) = b+ \mathbb{E}(X\mid X \geq a-b) =b + \frac{\int_{a-b}^\infty xf(x) dx}{\int_{a-b}^\infty f(x) dx}$$ and show that it is non negative. I computed $$g'(b)=1+ \frac{(a-b)f(a-b)\int_{a-b}^\infty f(x) dx-f(a-b)\int_{a-b}^\infty xf(x) dx}{\left(\int_{a-b}^\infty f(x) dx\right)^2}$$ but I couldn't show that the fraction is at least $-1$. I would appreciate any help.