Let $F$ and $G$ be the cumulative functions of $N(\mu_1,\sigma_1^2)$ and $N(\mu_2,\sigma_2^2)$. Let $U$ be a uniformly distributed random variable on $[0,1]$. How to calculate $E[(F^{-1}(U)-G^{-1}(U))^2]$, where $F^{-1}(t)=\inf\{x\in\Bbb{R}|F(x)>t\}$?
For standard normal random variables, $F^{-1}$ is named as probit function, but I cannot find a direct expression to calculate the expectation as an integration.