Is it possible to find a lower bound of this integral? $\displaystyle\int^A_0 (A-x)p(x)\ dx$. Here $p(x)$ is some probability distribution with known mean and standard deviation and $A$ is a constant.
I was trying to simplify this as $A\displaystyle\int^A_0 p(x)\ dx - \displaystyle\int^A_0 xp(x)\ dx $. The lower bound on the first integral can be found using Markov's inequality but how to find the upper bound of the second integral? Also, will this bound be tight?