Suppose that $(s \mid a) \sim N(a, \tau^2)$ with pdf $g(s \mid a)$. Then, for $s_1 > s_2$, I have trouble showing that $G(s_1 \mid a) \int_{s_2}^\infty s g(s \mid a) ds $ is increasing in $a \in (-\infty,p)$ and decreasing in $a \in [p, \infty)$, although my simulations show that it's true.
Can you please help me?
My thoughts so far: $G(s_1 \mid a)$ is decreasing in $a$ and $\int_{s_2}^\infty s g(s \mid a) ds $ is increasing. But the product is unclear.