Here $\alpha$ is a random variable and the expectation is taken with respect to that variable. I am wondering if it's the same in any case or there's a theorem quantifying how close both things are. I think I've come to proove that if $\dfrac{\partial f(p,\alpha)}{\partial p}$ never change sign in $\alpha$ the results are the same, according to the mean value theorem, but I'm looking for more general results.
Thank you