What is $\mathbb{E}\left( \Phi^{-1}(U)\right)$, $U \sim \mathcal{U}(0,1)$? Let $\Phi(\cdot)$ denote the CDF of a standard normal random variable and let $U \sim \mathcal{U}(0,1)$. What can we say about
$$\mathbb{E}\left( \Phi^{-1}(U)\right)?$$

Inverse CDF is a method to create random samples from given distributions. So, $\Phi^{-1}(U)$ is standard normal, and its mean is $0$.
