Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find
$$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find
$$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
HINT: If $X$ is standard normally distributed, then $\Phi(X)$ is uniformly distributed.
Since $\Phi$ is a strictly increasing function whose range is $(0,1)$, we have for $0
But if $u>1$ then $\Pr(\Phi(X)\ge u)$ is $0$ since that event is impossible.
That tells you what to integrate.