Variance of a Cumulative Distribution Function of Normal Distribution

Question

Suppose, $X\sim N(\mu,\sigma^2)$.

Can anyone help in finding the following : $\text{Var }\Phi(X)$ ?

Here, $\Phi(x)$ is the "Cumulative Distribution Function" of the above-mentioned normal distribution.

Thanking you in advance.

Answer 1

Some of the following conditions can be left out, but let's keep it easy.

Let $F$ as CDF of random variable $X$ be continuous, strictly increasing and taking values in $(0,1)$.

Then $F:\mathbb R\to (0,1)$ is a bijection, so has an inverse. For $u\in(0,1)$ we find:$$F(X)\leq u\iff X\leq F^{-1}(u)$$ hence:$$P(F(X)\leq u)= P(X\leq F^{-1}(u))=F(F^{-1}(u))=u$$

So what is the distribution of $F(X)$?

If you know the distribution then you can also find the corresponding variance.

Apply this on $F=\Phi$.