I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate $$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that allow an analytical solution. The next best thing is an approximation that is close for a wide range of $a$ and $b$, given that $G(t)<.01$, $\lambda(a(t+b))>.1$, and $\lambda'(a(t+b))<1$: that is, the integration is over the left tail of $G$, and $a$ and $b$ are such that the integrand is on the order of $\frac{1}{2}$ at the right edge of the range of integration and doesn't disappear too rapidly moving to the left.
