Accumulated-local (AL) profile for $X^1$

Another alternative to define the effect of the variable.

$$\begin{align} g_{AL}^{1}(z) &= \int_{-1}^z E \left[\frac{\partial f(X^1, X^2)}{\partial X^1} | X^1 = v \right] dv \nonumber \\ & = \int_{-1}^z E \left[X^2 | X^1 = v \right] dv = \int_{-1}^z v dv = (z^2 - 1)/2 \end{align}$$