I am reading a paper where there is an approximation of an expected value. I am not sure what sort of approximation method they are using. Reminds me a bit of Taylors Theorem, but I am just not connecting the dots.
\begin{align*} P(XC + \sqrt{X}A + B \leq z) = & E\left [I(XC + \sqrt{X}A + B \leq z)\right] \\ = & E\left [I(XC \leq z)\right] + E\left [I'\right]A + O(E(A^2)) + O(E|AB|) + O(EB) \\ \end{align*}
where
$I' = \frac{\partial}{\partial A} P(XC + \sqrt{X}A + B \leq Z) $ evaluated at $A= 0$, $B=0$, and $C=C$
$I(\cdot\leq Z)$ is the indicator function
$X, A, B, C$ all have random components