# Integrating inverse cumulative of standard Normal Distribution

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

## 1 Answer

Note that

$$\Phi^{-1}[\Phi(k)] = k$$

So subbing $u=\Phi(k)$, $du = \Phi'(k) dk = \phi(k) dk$, we get

$$\int_{1-p}^1 du \, \Phi^{-1}(u) = \int_{\Phi^{-1}(1-p)}^{\Phi^{-1}(1)} dk \, \Phi^{-1}[\Phi(k)] \Phi'(k) = \int_{\Phi^{-1}(1-p)}^{\infty} dk \, k\, \phi(k)$$

• Thank you! Much appreciated! I didn't think of the first thing you wrote! – Trelokoritso Feb 17 '16 at 9:40