Integrating inverse cumulative of standard Normal Distribution
By Emily Wilson •
I am studying this book and a particular line reads
$\int_{1-p}^1 \Phi^{-1}(u)du=${set $k=\Phi^{-1}(u)$}=$\int_{\Phi^{-1}(1-p)}^\infty k\phi(k)du$, where $\Phi$ and $\phi$ are the cumulative and density functions of the standard normal distribution.
and i cannot see how this is derived.
Does any kind person have the appetite to explain? Thanks in advance
$\endgroup$1 Answer
$\begingroup$Note that
$$\Phi^{-1}[\Phi(k)] = k$$
So subbing $u=\Phi(k)$, $du = \Phi'(k) dk = \phi(k) dk$, we get
$$\int_{1-p}^1 du \, \Phi^{-1}(u) = \int_{\Phi^{-1}(1-p)}^{\Phi^{-1}(1)} dk \, \Phi^{-1}[\Phi(k)] \Phi'(k) = \int_{\Phi^{-1}(1-p)}^{\infty} dk \, k\, \phi(k) $$
$\endgroup$ 1
