During a research project I was analyzing the shape of the conditional expectation of the Beta distribution, $g(t) = E[X\,|\, X>t]$ for $X\sim \mathrm{Beta}(\alpha,\beta)$.
Using numeric calculations, I got that $g(t)$ is linear when $\alpha = 1$ and concave for other values (both when $\alpha>\beta$ and vice versa). I was wondering if anyone has a reference or explanation for why this is so.
Thanks!