Suppose that $Z \sim \mathcal{N}(0,1)$ is a Gaussian random variable with mean zero and variance $1$. Let $F(z) \equiv \mathbb{P}(Z \leq z)$ be the cumulative distribution function $\operatorname{(CDF)}$ of $Z$. Define a new random variable $Y$ as $Y=F(Z)$. This means that the random variable $Y$ is obtained by evaluating the $\operatorname{CDF} F(\cdot)$ at randomly chosen points. Then the value of $\mathbb{E}[Y]$ is:
- $F(1)$
- $1$
- $\frac{1}{2}$
- $\frac{1}{\sqrt{2 \pi}}$
- $\frac{\pi}{4}$