I'm sorry if this has been already asked, i don't really know how to search for this (don't know key words).
My problem is the following.
I know this relation :
$$\phi (z) + \phi (-z) = 1$$ where $\phi$ is the cumulative function of a normal distribution.
In an exercice of my book of statistics, they are using what appears to be this formula :
$$ - \phi ^{-1} ( \alpha ) = \phi ^{-1} ( 1 - \alpha ) $$
why would this formula be true ? A plot verifies that this is true, but i was searching for a demonstration (even a small one hehe).
Thank you !
Let $x=\phi (z)$. Then $z=\phi^{-1}(x)$ and $x+\phi (-z)=1$. Hence $\phi (-z)=1-x$ which gives $-z=\phi^{-1}(1-x)$. Thus $\phi^{-1}(x)=z=-\phi^{-1}(1-x)$ which is what you want to prove.