Question-
Let $X$ follows $N(\mu,\sigma^2)$. Show that $\mathbb{E}(\Phi(X)) \neq \frac{1}{2}$ for any $\mu \neq 0$, where $\Phi(X)$ is the cdf of $N(0,1)$ distribution.
But the theorem of Probability Integral transformation says that cdf of any random variable follows $Uniform (0,1)$ distribution. So the normal cdf should not depend on $\mu$ and always follow $Uniform (0,1)$ distribution with mean $\frac{1}{2}$. Can you please point out what I am missing here?
Thanks in advance!